Properties

Label 28-6422e14-1.1-c1e14-0-0
Degree $28$
Conductor $2.029\times 10^{53}$
Sign $1$
Analytic cond. $8.69488\times 10^{23}$
Root an. cond. $7.16100$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 14·2-s + 4·3-s + 105·4-s + 2·5-s − 56·6-s − 2·7-s − 560·8-s − 4·9-s − 28·10-s − 10·11-s + 420·12-s + 28·14-s + 8·15-s + 2.38e3·16-s + 2·17-s + 56·18-s − 14·19-s + 210·20-s − 8·21-s + 140·22-s + 12·23-s − 2.24e3·24-s − 11·25-s − 38·27-s − 210·28-s + 4·29-s − 112·30-s + ⋯
L(s)  = 1  − 9.89·2-s + 2.30·3-s + 52.5·4-s + 0.894·5-s − 22.8·6-s − 0.755·7-s − 197.·8-s − 4/3·9-s − 8.85·10-s − 3.01·11-s + 121.·12-s + 7.48·14-s + 2.06·15-s + 595·16-s + 0.485·17-s + 13.1·18-s − 3.21·19-s + 46.9·20-s − 1.74·21-s + 29.8·22-s + 2.50·23-s − 457.·24-s − 2.19·25-s − 7.31·27-s − 39.6·28-s + 0.742·29-s − 20.4·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{14} \cdot 13^{28} \cdot 19^{14}\right)^{s/2} \, \Gamma_{\C}(s)^{14} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{14} \cdot 13^{28} \cdot 19^{14}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{14} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(28\)
Conductor: \(2^{14} \cdot 13^{28} \cdot 19^{14}\)
Sign: $1$
Analytic conductor: \(8.69488\times 10^{23}\)
Root analytic conductor: \(7.16100\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((28,\ 2^{14} \cdot 13^{28} \cdot 19^{14} ,\ ( \ : [1/2]^{14} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.008011867245\)
\(L(\frac12)\) \(\approx\) \(0.008011867245\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 + T )^{14} \)
13 \( 1 \)
19 \( ( 1 + T )^{14} \)
good3 \( 1 - 4 T + 20 T^{2} - 58 T^{3} + 191 T^{4} - 2 p^{5} T^{5} + 1306 T^{6} - 334 p^{2} T^{7} + 7028 T^{8} - 14836 T^{9} + 31220 T^{10} - 61076 T^{11} + 117701 T^{12} - 213952 T^{13} + 380584 T^{14} - 213952 p T^{15} + 117701 p^{2} T^{16} - 61076 p^{3} T^{17} + 31220 p^{4} T^{18} - 14836 p^{5} T^{19} + 7028 p^{6} T^{20} - 334 p^{9} T^{21} + 1306 p^{8} T^{22} - 2 p^{14} T^{23} + 191 p^{10} T^{24} - 58 p^{11} T^{25} + 20 p^{12} T^{26} - 4 p^{13} T^{27} + p^{14} T^{28} \)
5 \( 1 - 2 T + 3 p T^{2} - 42 T^{3} + 44 p T^{4} - 576 T^{5} + 2194 T^{6} - 5716 T^{7} + 19323 T^{8} - 46434 T^{9} + 137927 T^{10} - 315362 T^{11} + 861096 T^{12} - 1836028 T^{13} + 4588368 T^{14} - 1836028 p T^{15} + 861096 p^{2} T^{16} - 315362 p^{3} T^{17} + 137927 p^{4} T^{18} - 46434 p^{5} T^{19} + 19323 p^{6} T^{20} - 5716 p^{7} T^{21} + 2194 p^{8} T^{22} - 576 p^{9} T^{23} + 44 p^{11} T^{24} - 42 p^{11} T^{25} + 3 p^{13} T^{26} - 2 p^{13} T^{27} + p^{14} T^{28} \)
7 \( 1 + 2 T + 37 T^{2} + 52 T^{3} + 692 T^{4} + 68 p T^{5} + 8723 T^{6} - 20 T^{7} + 89015 T^{8} - 43350 T^{9} + 837364 T^{10} - 511212 T^{11} + 7327009 T^{12} - 552492 p T^{13} + 55846718 T^{14} - 552492 p^{2} T^{15} + 7327009 p^{2} T^{16} - 511212 p^{3} T^{17} + 837364 p^{4} T^{18} - 43350 p^{5} T^{19} + 89015 p^{6} T^{20} - 20 p^{7} T^{21} + 8723 p^{8} T^{22} + 68 p^{10} T^{23} + 692 p^{10} T^{24} + 52 p^{11} T^{25} + 37 p^{12} T^{26} + 2 p^{13} T^{27} + p^{14} T^{28} \)
11 \( 1 + 10 T + 84 T^{2} + 508 T^{3} + 2900 T^{4} + 14176 T^{5} + 68405 T^{6} + 301338 T^{7} + 1305420 T^{8} + 5227930 T^{9} + 20633764 T^{10} + 6946912 p T^{11} + 279713911 T^{12} + 968254618 T^{13} + 3307838342 T^{14} + 968254618 p T^{15} + 279713911 p^{2} T^{16} + 6946912 p^{4} T^{17} + 20633764 p^{4} T^{18} + 5227930 p^{5} T^{19} + 1305420 p^{6} T^{20} + 301338 p^{7} T^{21} + 68405 p^{8} T^{22} + 14176 p^{9} T^{23} + 2900 p^{10} T^{24} + 508 p^{11} T^{25} + 84 p^{12} T^{26} + 10 p^{13} T^{27} + p^{14} T^{28} \)
17 \( 1 - 2 T + 125 T^{2} - 288 T^{3} + 7950 T^{4} - 19292 T^{5} + 338218 T^{6} - 817376 T^{7} + 10775317 T^{8} - 24937068 T^{9} + 274372361 T^{10} - 594319866 T^{11} + 5833380620 T^{12} - 11749283996 T^{13} + 106490754304 T^{14} - 11749283996 p T^{15} + 5833380620 p^{2} T^{16} - 594319866 p^{3} T^{17} + 274372361 p^{4} T^{18} - 24937068 p^{5} T^{19} + 10775317 p^{6} T^{20} - 817376 p^{7} T^{21} + 338218 p^{8} T^{22} - 19292 p^{9} T^{23} + 7950 p^{10} T^{24} - 288 p^{11} T^{25} + 125 p^{12} T^{26} - 2 p^{13} T^{27} + p^{14} T^{28} \)
23 \( 1 - 12 T + 214 T^{2} - 2066 T^{3} + 21281 T^{4} - 170144 T^{5} + 57678 p T^{6} - 9019462 T^{7} + 58994228 T^{8} - 351068156 T^{9} + 2028003514 T^{10} - 10898667480 T^{11} + 57401410503 T^{12} - 286278879920 T^{13} + 61101164302 p T^{14} - 286278879920 p T^{15} + 57401410503 p^{2} T^{16} - 10898667480 p^{3} T^{17} + 2028003514 p^{4} T^{18} - 351068156 p^{5} T^{19} + 58994228 p^{6} T^{20} - 9019462 p^{7} T^{21} + 57678 p^{9} T^{22} - 170144 p^{9} T^{23} + 21281 p^{10} T^{24} - 2066 p^{11} T^{25} + 214 p^{12} T^{26} - 12 p^{13} T^{27} + p^{14} T^{28} \)
29 \( 1 - 4 T + 189 T^{2} - 746 T^{3} + 18697 T^{4} - 75410 T^{5} + 1283138 T^{6} - 5231392 T^{7} + 68302584 T^{8} - 274490690 T^{9} + 2984711890 T^{10} - 11501393832 T^{11} + 109967689103 T^{12} - 397609147698 T^{13} + 3451127078525 T^{14} - 397609147698 p T^{15} + 109967689103 p^{2} T^{16} - 11501393832 p^{3} T^{17} + 2984711890 p^{4} T^{18} - 274490690 p^{5} T^{19} + 68302584 p^{6} T^{20} - 5231392 p^{7} T^{21} + 1283138 p^{8} T^{22} - 75410 p^{9} T^{23} + 18697 p^{10} T^{24} - 746 p^{11} T^{25} + 189 p^{12} T^{26} - 4 p^{13} T^{27} + p^{14} T^{28} \)
31 \( 1 + 4 T + 241 T^{2} + 936 T^{3} + 28725 T^{4} + 102110 T^{5} + 2216579 T^{6} + 6846774 T^{7} + 123958383 T^{8} + 316564780 T^{9} + 5409788435 T^{10} + 11072468584 T^{11} + 197263174043 T^{12} + 336356123896 T^{13} + 6395747675250 T^{14} + 336356123896 p T^{15} + 197263174043 p^{2} T^{16} + 11072468584 p^{3} T^{17} + 5409788435 p^{4} T^{18} + 316564780 p^{5} T^{19} + 123958383 p^{6} T^{20} + 6846774 p^{7} T^{21} + 2216579 p^{8} T^{22} + 102110 p^{9} T^{23} + 28725 p^{10} T^{24} + 936 p^{11} T^{25} + 241 p^{12} T^{26} + 4 p^{13} T^{27} + p^{14} T^{28} \)
37 \( 1 + 18 T + 10 p T^{2} + 4572 T^{3} + 56401 T^{4} + 14684 p T^{5} + 5083304 T^{6} + 40993530 T^{7} + 320678976 T^{8} + 2282580942 T^{9} + 15904341944 T^{10} + 104343415908 T^{11} + 677230087126 T^{12} + 4215795440290 T^{13} + 26115145423932 T^{14} + 4215795440290 p T^{15} + 677230087126 p^{2} T^{16} + 104343415908 p^{3} T^{17} + 15904341944 p^{4} T^{18} + 2282580942 p^{5} T^{19} + 320678976 p^{6} T^{20} + 40993530 p^{7} T^{21} + 5083304 p^{8} T^{22} + 14684 p^{10} T^{23} + 56401 p^{10} T^{24} + 4572 p^{11} T^{25} + 10 p^{13} T^{26} + 18 p^{13} T^{27} + p^{14} T^{28} \)
41 \( 1 + 6 T + 250 T^{2} + 1614 T^{3} + 33110 T^{4} + 202264 T^{5} + 3028515 T^{6} + 16488356 T^{7} + 209212193 T^{8} + 1012377750 T^{9} + 11638491348 T^{10} + 50769917398 T^{11} + 555663618152 T^{12} + 2231526511620 T^{13} + 23843898720030 T^{14} + 2231526511620 p T^{15} + 555663618152 p^{2} T^{16} + 50769917398 p^{3} T^{17} + 11638491348 p^{4} T^{18} + 1012377750 p^{5} T^{19} + 209212193 p^{6} T^{20} + 16488356 p^{7} T^{21} + 3028515 p^{8} T^{22} + 202264 p^{9} T^{23} + 33110 p^{10} T^{24} + 1614 p^{11} T^{25} + 250 p^{12} T^{26} + 6 p^{13} T^{27} + p^{14} T^{28} \)
43 \( 1 - 28 T + 557 T^{2} - 7700 T^{3} + 90346 T^{4} - 883072 T^{5} + 7963981 T^{6} - 65243564 T^{7} + 524691940 T^{8} - 4021726320 T^{9} + 30865402519 T^{10} - 225888812568 T^{11} + 1634976978249 T^{12} - 11193536806572 T^{13} + 75457826587358 T^{14} - 11193536806572 p T^{15} + 1634976978249 p^{2} T^{16} - 225888812568 p^{3} T^{17} + 30865402519 p^{4} T^{18} - 4021726320 p^{5} T^{19} + 524691940 p^{6} T^{20} - 65243564 p^{7} T^{21} + 7963981 p^{8} T^{22} - 883072 p^{9} T^{23} + 90346 p^{10} T^{24} - 7700 p^{11} T^{25} + 557 p^{12} T^{26} - 28 p^{13} T^{27} + p^{14} T^{28} \)
47 \( 1 - 20 T + 568 T^{2} - 8204 T^{3} + 135835 T^{4} - 1555080 T^{5} + 19038707 T^{6} - 181470614 T^{7} + 1798332629 T^{8} - 14786666372 T^{9} + 125364606057 T^{10} - 921986116410 T^{11} + 7031697897258 T^{12} - 48166484335366 T^{13} + 345806132834576 T^{14} - 48166484335366 p T^{15} + 7031697897258 p^{2} T^{16} - 921986116410 p^{3} T^{17} + 125364606057 p^{4} T^{18} - 14786666372 p^{5} T^{19} + 1798332629 p^{6} T^{20} - 181470614 p^{7} T^{21} + 19038707 p^{8} T^{22} - 1555080 p^{9} T^{23} + 135835 p^{10} T^{24} - 8204 p^{11} T^{25} + 568 p^{12} T^{26} - 20 p^{13} T^{27} + p^{14} T^{28} \)
53 \( 1 - 12 T + 8 p T^{2} - 3736 T^{3} + 80247 T^{4} - 553834 T^{5} + 9607528 T^{6} - 53365582 T^{7} + 15946403 p T^{8} - 3800278134 T^{9} + 59586601870 T^{10} - 219711873404 T^{11} + 3609670170903 T^{12} - 11549850773718 T^{13} + 198559380171045 T^{14} - 11549850773718 p T^{15} + 3609670170903 p^{2} T^{16} - 219711873404 p^{3} T^{17} + 59586601870 p^{4} T^{18} - 3800278134 p^{5} T^{19} + 15946403 p^{7} T^{20} - 53365582 p^{7} T^{21} + 9607528 p^{8} T^{22} - 553834 p^{9} T^{23} + 80247 p^{10} T^{24} - 3736 p^{11} T^{25} + 8 p^{13} T^{26} - 12 p^{13} T^{27} + p^{14} T^{28} \)
59 \( 1 + 16 T + 642 T^{2} + 8770 T^{3} + 195911 T^{4} + 2348644 T^{5} + 38001368 T^{6} + 406247376 T^{7} + 5264518758 T^{8} + 50628670078 T^{9} + 553250441830 T^{10} + 4802813877956 T^{11} + 45617940886651 T^{12} + 357015496218586 T^{13} + 3003614058219260 T^{14} + 357015496218586 p T^{15} + 45617940886651 p^{2} T^{16} + 4802813877956 p^{3} T^{17} + 553250441830 p^{4} T^{18} + 50628670078 p^{5} T^{19} + 5264518758 p^{6} T^{20} + 406247376 p^{7} T^{21} + 38001368 p^{8} T^{22} + 2348644 p^{9} T^{23} + 195911 p^{10} T^{24} + 8770 p^{11} T^{25} + 642 p^{12} T^{26} + 16 p^{13} T^{27} + p^{14} T^{28} \)
61 \( 1 - 30 T + 769 T^{2} - 13588 T^{3} + 216816 T^{4} - 2877954 T^{5} + 35509244 T^{6} - 388094162 T^{7} + 4017335991 T^{8} - 38016531312 T^{9} + 346609016089 T^{10} - 2956962910022 T^{11} + 24766537955720 T^{12} - 197913023858548 T^{13} + 1575766262984660 T^{14} - 197913023858548 p T^{15} + 24766537955720 p^{2} T^{16} - 2956962910022 p^{3} T^{17} + 346609016089 p^{4} T^{18} - 38016531312 p^{5} T^{19} + 4017335991 p^{6} T^{20} - 388094162 p^{7} T^{21} + 35509244 p^{8} T^{22} - 2877954 p^{9} T^{23} + 216816 p^{10} T^{24} - 13588 p^{11} T^{25} + 769 p^{12} T^{26} - 30 p^{13} T^{27} + p^{14} T^{28} \)
67 \( 1 + 2 T + 622 T^{2} + 868 T^{3} + 189772 T^{4} + 172970 T^{5} + 37785149 T^{6} + 20206792 T^{7} + 5502461396 T^{8} + 1431417506 T^{9} + 621728042986 T^{10} + 52778209336 T^{11} + 56345047161479 T^{12} + 193132191270 T^{13} + 4167432376226486 T^{14} + 193132191270 p T^{15} + 56345047161479 p^{2} T^{16} + 52778209336 p^{3} T^{17} + 621728042986 p^{4} T^{18} + 1431417506 p^{5} T^{19} + 5502461396 p^{6} T^{20} + 20206792 p^{7} T^{21} + 37785149 p^{8} T^{22} + 172970 p^{9} T^{23} + 189772 p^{10} T^{24} + 868 p^{11} T^{25} + 622 p^{12} T^{26} + 2 p^{13} T^{27} + p^{14} T^{28} \)
71 \( 1 + 44 T + 1527 T^{2} + 37854 T^{3} + 808892 T^{4} + 14637902 T^{5} + 237835767 T^{6} + 686168 p^{2} T^{7} + 46168999184 T^{8} + 565651683512 T^{9} + 6440396707825 T^{10} + 68179687846142 T^{11} + 675450791600291 T^{12} + 6261311413432994 T^{13} + 54488939020525122 T^{14} + 6261311413432994 p T^{15} + 675450791600291 p^{2} T^{16} + 68179687846142 p^{3} T^{17} + 6440396707825 p^{4} T^{18} + 565651683512 p^{5} T^{19} + 46168999184 p^{6} T^{20} + 686168 p^{9} T^{21} + 237835767 p^{8} T^{22} + 14637902 p^{9} T^{23} + 808892 p^{10} T^{24} + 37854 p^{11} T^{25} + 1527 p^{12} T^{26} + 44 p^{13} T^{27} + p^{14} T^{28} \)
73 \( 1 - 36 T + 1248 T^{2} - 29600 T^{3} + 637894 T^{4} - 11503588 T^{5} + 190198015 T^{6} - 2795246592 T^{7} + 38101738871 T^{8} - 473867184252 T^{9} + 5509919210434 T^{10} - 59230820249640 T^{11} + 598190014243967 T^{12} - 5621180568381624 T^{13} + 49742057757391257 T^{14} - 5621180568381624 p T^{15} + 598190014243967 p^{2} T^{16} - 59230820249640 p^{3} T^{17} + 5509919210434 p^{4} T^{18} - 473867184252 p^{5} T^{19} + 38101738871 p^{6} T^{20} - 2795246592 p^{7} T^{21} + 190198015 p^{8} T^{22} - 11503588 p^{9} T^{23} + 637894 p^{10} T^{24} - 29600 p^{11} T^{25} + 1248 p^{12} T^{26} - 36 p^{13} T^{27} + p^{14} T^{28} \)
79 \( 1 - 34 T + 1176 T^{2} - 26146 T^{3} + 552860 T^{4} - 9433460 T^{5} + 152584761 T^{6} - 2151456794 T^{7} + 28914582640 T^{8} - 351004144928 T^{9} + 4084216339840 T^{10} - 43712593413592 T^{11} + 450197246339987 T^{12} - 4302677194949406 T^{13} + 39646676959150046 T^{14} - 4302677194949406 p T^{15} + 450197246339987 p^{2} T^{16} - 43712593413592 p^{3} T^{17} + 4084216339840 p^{4} T^{18} - 351004144928 p^{5} T^{19} + 28914582640 p^{6} T^{20} - 2151456794 p^{7} T^{21} + 152584761 p^{8} T^{22} - 9433460 p^{9} T^{23} + 552860 p^{10} T^{24} - 26146 p^{11} T^{25} + 1176 p^{12} T^{26} - 34 p^{13} T^{27} + p^{14} T^{28} \)
83 \( 1 + 2 T + 481 T^{2} - 1258 T^{3} + 111749 T^{4} - 794024 T^{5} + 19000799 T^{6} - 193229162 T^{7} + 2795910017 T^{8} - 30712186268 T^{9} + 355025122263 T^{10} - 3781004904544 T^{11} + 37880120693129 T^{12} - 380665058904802 T^{13} + 3410457866926418 T^{14} - 380665058904802 p T^{15} + 37880120693129 p^{2} T^{16} - 3781004904544 p^{3} T^{17} + 355025122263 p^{4} T^{18} - 30712186268 p^{5} T^{19} + 2795910017 p^{6} T^{20} - 193229162 p^{7} T^{21} + 19000799 p^{8} T^{22} - 794024 p^{9} T^{23} + 111749 p^{10} T^{24} - 1258 p^{11} T^{25} + 481 p^{12} T^{26} + 2 p^{13} T^{27} + p^{14} T^{28} \)
89 \( 1 + 42 T + 1593 T^{2} + 41936 T^{3} + 993937 T^{4} + 19736304 T^{5} + 359135691 T^{6} + 5808335060 T^{7} + 87209038355 T^{8} + 1194457208864 T^{9} + 15313897036891 T^{10} + 181455580095158 T^{11} + 2022398357789019 T^{12} + 20964891685313436 T^{13} + 204883026258954690 T^{14} + 20964891685313436 p T^{15} + 2022398357789019 p^{2} T^{16} + 181455580095158 p^{3} T^{17} + 15313897036891 p^{4} T^{18} + 1194457208864 p^{5} T^{19} + 87209038355 p^{6} T^{20} + 5808335060 p^{7} T^{21} + 359135691 p^{8} T^{22} + 19736304 p^{9} T^{23} + 993937 p^{10} T^{24} + 41936 p^{11} T^{25} + 1593 p^{12} T^{26} + 42 p^{13} T^{27} + p^{14} T^{28} \)
97 \( 1 + 40 T + 1250 T^{2} + 25448 T^{3} + 452443 T^{4} + 6289104 T^{5} + 83353972 T^{6} + 956222160 T^{7} + 11650677545 T^{8} + 129911125144 T^{9} + 1555111218494 T^{10} + 16350784238808 T^{11} + 178755380165627 T^{12} + 1715719482838880 T^{13} + 17771700065183448 T^{14} + 1715719482838880 p T^{15} + 178755380165627 p^{2} T^{16} + 16350784238808 p^{3} T^{17} + 1555111218494 p^{4} T^{18} + 129911125144 p^{5} T^{19} + 11650677545 p^{6} T^{20} + 956222160 p^{7} T^{21} + 83353972 p^{8} T^{22} + 6289104 p^{9} T^{23} + 452443 p^{10} T^{24} + 25448 p^{11} T^{25} + 1250 p^{12} T^{26} + 40 p^{13} T^{27} + p^{14} T^{28} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{28} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−1.93592738451580045795449363492, −1.91189063071213877190237703361, −1.89167259767521213923247260777, −1.82437828367317018401114200100, −1.78216288957183871601661157544, −1.77557669515231980750712242556, −1.76300045398413865538094256980, −1.59757209672983397690166097632, −1.40975765562927080981490820423, −1.35600403370868827244656972751, −1.32477879861516692624576058346, −1.24390655692505025301136440352, −1.22002075720581835491674826680, −0.992152768620149376161941001458, −0.77245776205247139677564933178, −0.77158857978815563435334094327, −0.76672131849248544570044114829, −0.64459711574933456383513639045, −0.60733503273757200726146235666, −0.46882295836085496098871313949, −0.43685598825850214943889607098, −0.41289113158626736094046493470, −0.26234986157726880472889175399, −0.15746067848677417523477551420, −0.05198208272884298449074385928, 0.05198208272884298449074385928, 0.15746067848677417523477551420, 0.26234986157726880472889175399, 0.41289113158626736094046493470, 0.43685598825850214943889607098, 0.46882295836085496098871313949, 0.60733503273757200726146235666, 0.64459711574933456383513639045, 0.76672131849248544570044114829, 0.77158857978815563435334094327, 0.77245776205247139677564933178, 0.992152768620149376161941001458, 1.22002075720581835491674826680, 1.24390655692505025301136440352, 1.32477879861516692624576058346, 1.35600403370868827244656972751, 1.40975765562927080981490820423, 1.59757209672983397690166097632, 1.76300045398413865538094256980, 1.77557669515231980750712242556, 1.78216288957183871601661157544, 1.82437828367317018401114200100, 1.89167259767521213923247260777, 1.91189063071213877190237703361, 1.93592738451580045795449363492

Graph of the $Z$-function along the critical line

Plot not available for L-functions of degree greater than 10.