Properties

Degree $28$
Conductor $4.667\times 10^{54}$
Sign $1$
Motivic weight $1$
Primitive no
Self-dual yes
Analytic rank $14$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 14·2-s + 14·3-s + 105·4-s − 6·5-s − 196·6-s − 4·7-s − 560·8-s + 105·9-s + 84·10-s − 8·11-s + 1.47e3·12-s − 14·13-s + 56·14-s − 84·15-s + 2.38e3·16-s − 4·17-s − 1.47e3·18-s − 19-s − 630·20-s − 56·21-s + 112·22-s − 9·23-s − 7.84e3·24-s − 5·25-s + 196·26-s + 560·27-s − 420·28-s + ⋯
L(s)  = 1  − 9.89·2-s + 8.08·3-s + 52.5·4-s − 2.68·5-s − 80.0·6-s − 1.51·7-s − 197.·8-s + 35·9-s + 26.5·10-s − 2.41·11-s + 424.·12-s − 3.88·13-s + 14.9·14-s − 21.6·15-s + 595·16-s − 0.970·17-s − 346.·18-s − 0.229·19-s − 140.·20-s − 12.2·21-s + 23.8·22-s − 1.87·23-s − 1.60e3·24-s − 25-s + 38.4·26-s + 107.·27-s − 79.3·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{14} \cdot 3^{14} \cdot 13^{14} \cdot 103^{14}\right)^{s/2} \, \Gamma_{\C}(s)^{14} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{14} \cdot 3^{14} \cdot 13^{14} \cdot 103^{14}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{14} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(28\)
Conductor: \(2^{14} \cdot 3^{14} \cdot 13^{14} \cdot 103^{14}\)
Sign: $1$
Motivic weight: \(1\)
Character: induced by $\chi_{8034} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(14\)
Selberg data: \((28,\ 2^{14} \cdot 3^{14} \cdot 13^{14} \cdot 103^{14} ,\ ( \ : [1/2]^{14} ),\ 1 )\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 + T )^{14} \)
3 \( ( 1 - T )^{14} \)
13 \( ( 1 + T )^{14} \)
103 \( ( 1 - T )^{14} \)
good5 \( 1 + 6 T + 41 T^{2} + 183 T^{3} + 804 T^{4} + 2916 T^{5} + 10253 T^{6} + 32069 T^{7} + 97548 T^{8} + 54476 p T^{9} + 148707 p T^{10} + 1893519 T^{11} + 4723591 T^{12} + 11092331 T^{13} + 25527862 T^{14} + 11092331 p T^{15} + 4723591 p^{2} T^{16} + 1893519 p^{3} T^{17} + 148707 p^{5} T^{18} + 54476 p^{6} T^{19} + 97548 p^{6} T^{20} + 32069 p^{7} T^{21} + 10253 p^{8} T^{22} + 2916 p^{9} T^{23} + 804 p^{10} T^{24} + 183 p^{11} T^{25} + 41 p^{12} T^{26} + 6 p^{13} T^{27} + p^{14} T^{28} \)
7 \( 1 + 4 T + 45 T^{2} + 150 T^{3} + 970 T^{4} + 390 p T^{5} + 1909 p T^{6} + 32257 T^{7} + 134875 T^{8} + 283916 T^{9} + 1104598 T^{10} + 2080278 T^{11} + 8060734 T^{12} + 290707 p^{2} T^{13} + 56608556 T^{14} + 290707 p^{3} T^{15} + 8060734 p^{2} T^{16} + 2080278 p^{3} T^{17} + 1104598 p^{4} T^{18} + 283916 p^{5} T^{19} + 134875 p^{6} T^{20} + 32257 p^{7} T^{21} + 1909 p^{9} T^{22} + 390 p^{10} T^{23} + 970 p^{10} T^{24} + 150 p^{11} T^{25} + 45 p^{12} T^{26} + 4 p^{13} T^{27} + p^{14} T^{28} \)
11 \( 1 + 8 T + 115 T^{2} + 63 p T^{3} + 5771 T^{4} + 28541 T^{5} + 177373 T^{6} + 755945 T^{7} + 3867618 T^{8} + 14680639 T^{9} + 65285057 T^{10} + 226078711 T^{11} + 905329918 T^{12} + 2903978747 T^{13} + 10710168638 T^{14} + 2903978747 p T^{15} + 905329918 p^{2} T^{16} + 226078711 p^{3} T^{17} + 65285057 p^{4} T^{18} + 14680639 p^{5} T^{19} + 3867618 p^{6} T^{20} + 755945 p^{7} T^{21} + 177373 p^{8} T^{22} + 28541 p^{9} T^{23} + 5771 p^{10} T^{24} + 63 p^{12} T^{25} + 115 p^{12} T^{26} + 8 p^{13} T^{27} + p^{14} T^{28} \)
17 \( 1 + 4 T + 112 T^{2} + 455 T^{3} + 6661 T^{4} + 27062 T^{5} + 277881 T^{6} + 1107307 T^{7} + 8993960 T^{8} + 34454734 T^{9} + 236670370 T^{10} + 857297371 T^{11} + 5189048050 T^{12} + 17526492347 T^{13} + 95917900138 T^{14} + 17526492347 p T^{15} + 5189048050 p^{2} T^{16} + 857297371 p^{3} T^{17} + 236670370 p^{4} T^{18} + 34454734 p^{5} T^{19} + 8993960 p^{6} T^{20} + 1107307 p^{7} T^{21} + 277881 p^{8} T^{22} + 27062 p^{9} T^{23} + 6661 p^{10} T^{24} + 455 p^{11} T^{25} + 112 p^{12} T^{26} + 4 p^{13} T^{27} + p^{14} T^{28} \)
19 \( 1 + T + 106 T^{2} - 17 T^{3} + 294 p T^{4} - 6827 T^{5} + 211256 T^{6} - 402664 T^{7} + 6693371 T^{8} - 14345340 T^{9} + 182347476 T^{10} - 400413994 T^{11} + 4240436866 T^{12} - 9324043833 T^{13} + 85718030132 T^{14} - 9324043833 p T^{15} + 4240436866 p^{2} T^{16} - 400413994 p^{3} T^{17} + 182347476 p^{4} T^{18} - 14345340 p^{5} T^{19} + 6693371 p^{6} T^{20} - 402664 p^{7} T^{21} + 211256 p^{8} T^{22} - 6827 p^{9} T^{23} + 294 p^{11} T^{24} - 17 p^{11} T^{25} + 106 p^{12} T^{26} + p^{13} T^{27} + p^{14} T^{28} \)
23 \( 1 + 9 T + 218 T^{2} + 1394 T^{3} + 19904 T^{4} + 92422 T^{5} + 1060630 T^{6} + 3513688 T^{7} + 39696593 T^{8} + 92499915 T^{9} + 1227382805 T^{10} + 2173065758 T^{11} + 34402667030 T^{12} + 53128614126 T^{13} + 854793008294 T^{14} + 53128614126 p T^{15} + 34402667030 p^{2} T^{16} + 2173065758 p^{3} T^{17} + 1227382805 p^{4} T^{18} + 92499915 p^{5} T^{19} + 39696593 p^{6} T^{20} + 3513688 p^{7} T^{21} + 1060630 p^{8} T^{22} + 92422 p^{9} T^{23} + 19904 p^{10} T^{24} + 1394 p^{11} T^{25} + 218 p^{12} T^{26} + 9 p^{13} T^{27} + p^{14} T^{28} \)
29 \( 1 + 10 T + 276 T^{2} + 2229 T^{3} + 36268 T^{4} + 252689 T^{5} + 3101986 T^{6} + 19180659 T^{7} + 194296543 T^{8} + 1081823807 T^{9} + 326274194 p T^{10} + 47796897838 T^{11} + 370298779492 T^{12} + 1700390057896 T^{13} + 11845573980640 T^{14} + 1700390057896 p T^{15} + 370298779492 p^{2} T^{16} + 47796897838 p^{3} T^{17} + 326274194 p^{5} T^{18} + 1081823807 p^{5} T^{19} + 194296543 p^{6} T^{20} + 19180659 p^{7} T^{21} + 3101986 p^{8} T^{22} + 252689 p^{9} T^{23} + 36268 p^{10} T^{24} + 2229 p^{11} T^{25} + 276 p^{12} T^{26} + 10 p^{13} T^{27} + p^{14} T^{28} \)
31 \( 1 + 5 T + 246 T^{2} + 1015 T^{3} + 29204 T^{4} + 93459 T^{5} + 2215336 T^{6} + 4982146 T^{7} + 121083335 T^{8} + 156826740 T^{9} + 5165689838 T^{10} + 2248011146 T^{11} + 185126820172 T^{12} - 20406816505 T^{13} + 5962515707672 T^{14} - 20406816505 p T^{15} + 185126820172 p^{2} T^{16} + 2248011146 p^{3} T^{17} + 5165689838 p^{4} T^{18} + 156826740 p^{5} T^{19} + 121083335 p^{6} T^{20} + 4982146 p^{7} T^{21} + 2215336 p^{8} T^{22} + 93459 p^{9} T^{23} + 29204 p^{10} T^{24} + 1015 p^{11} T^{25} + 246 p^{12} T^{26} + 5 p^{13} T^{27} + p^{14} T^{28} \)
37 \( 1 + 4 T + 265 T^{2} + 469 T^{3} + 34658 T^{4} + 11966 T^{5} + 3187605 T^{6} - 1583648 T^{7} + 229272679 T^{8} - 225871475 T^{9} + 13348935918 T^{10} - 16395454920 T^{11} + 644409391740 T^{12} - 813518143628 T^{13} + 26053793698036 T^{14} - 813518143628 p T^{15} + 644409391740 p^{2} T^{16} - 16395454920 p^{3} T^{17} + 13348935918 p^{4} T^{18} - 225871475 p^{5} T^{19} + 229272679 p^{6} T^{20} - 1583648 p^{7} T^{21} + 3187605 p^{8} T^{22} + 11966 p^{9} T^{23} + 34658 p^{10} T^{24} + 469 p^{11} T^{25} + 265 p^{12} T^{26} + 4 p^{13} T^{27} + p^{14} T^{28} \)
41 \( 1 + 24 T + 430 T^{2} + 4986 T^{3} + 50235 T^{4} + 414972 T^{5} + 3441888 T^{6} + 26808105 T^{7} + 209107874 T^{8} + 35882838 p T^{9} + 9760386156 T^{10} + 62059530618 T^{11} + 405109179442 T^{12} + 2730537791825 T^{13} + 434777633260 p T^{14} + 2730537791825 p T^{15} + 405109179442 p^{2} T^{16} + 62059530618 p^{3} T^{17} + 9760386156 p^{4} T^{18} + 35882838 p^{6} T^{19} + 209107874 p^{6} T^{20} + 26808105 p^{7} T^{21} + 3441888 p^{8} T^{22} + 414972 p^{9} T^{23} + 50235 p^{10} T^{24} + 4986 p^{11} T^{25} + 430 p^{12} T^{26} + 24 p^{13} T^{27} + p^{14} T^{28} \)
43 \( 1 + 371 T^{2} + 72 T^{3} + 63145 T^{4} + 33658 T^{5} + 149859 p T^{6} + 6995121 T^{7} + 433134162 T^{8} + 870799568 T^{9} + 20088493843 T^{10} + 72850576834 T^{11} + 701555046604 T^{12} + 4321181093061 T^{13} + 25278722069618 T^{14} + 4321181093061 p T^{15} + 701555046604 p^{2} T^{16} + 72850576834 p^{3} T^{17} + 20088493843 p^{4} T^{18} + 870799568 p^{5} T^{19} + 433134162 p^{6} T^{20} + 6995121 p^{7} T^{21} + 149859 p^{9} T^{22} + 33658 p^{9} T^{23} + 63145 p^{10} T^{24} + 72 p^{11} T^{25} + 371 p^{12} T^{26} + p^{14} T^{28} \)
47 \( 1 + 32 T + 937 T^{2} + 18332 T^{3} + 325258 T^{4} + 4713494 T^{5} + 62869397 T^{6} + 732375533 T^{7} + 7957211532 T^{8} + 77953219786 T^{9} + 719664123905 T^{10} + 6098481944010 T^{11} + 49065402157493 T^{12} + 365915276110075 T^{13} + 2601395262021930 T^{14} + 365915276110075 p T^{15} + 49065402157493 p^{2} T^{16} + 6098481944010 p^{3} T^{17} + 719664123905 p^{4} T^{18} + 77953219786 p^{5} T^{19} + 7957211532 p^{6} T^{20} + 732375533 p^{7} T^{21} + 62869397 p^{8} T^{22} + 4713494 p^{9} T^{23} + 325258 p^{10} T^{24} + 18332 p^{11} T^{25} + 937 p^{12} T^{26} + 32 p^{13} T^{27} + p^{14} T^{28} \)
53 \( 1 + 5 T + 310 T^{2} + 514 T^{3} + 45257 T^{4} - 53879 T^{5} + 4716903 T^{6} - 15081766 T^{7} + 425253352 T^{8} - 1670614529 T^{9} + 33658762096 T^{10} - 134303865386 T^{11} + 2251304821622 T^{12} - 8826593113975 T^{13} + 128261303595494 T^{14} - 8826593113975 p T^{15} + 2251304821622 p^{2} T^{16} - 134303865386 p^{3} T^{17} + 33658762096 p^{4} T^{18} - 1670614529 p^{5} T^{19} + 425253352 p^{6} T^{20} - 15081766 p^{7} T^{21} + 4716903 p^{8} T^{22} - 53879 p^{9} T^{23} + 45257 p^{10} T^{24} + 514 p^{11} T^{25} + 310 p^{12} T^{26} + 5 p^{13} T^{27} + p^{14} T^{28} \)
59 \( 1 + 13 T + 5 p T^{2} + 2438 T^{3} + 43865 T^{4} + 332795 T^{5} + 5032470 T^{6} + 32139538 T^{7} + 443875251 T^{8} + 2666335917 T^{9} + 34772734577 T^{10} + 188346686888 T^{11} + 2323971058803 T^{12} + 12131936012011 T^{13} + 145893820387140 T^{14} + 12131936012011 p T^{15} + 2323971058803 p^{2} T^{16} + 188346686888 p^{3} T^{17} + 34772734577 p^{4} T^{18} + 2666335917 p^{5} T^{19} + 443875251 p^{6} T^{20} + 32139538 p^{7} T^{21} + 5032470 p^{8} T^{22} + 332795 p^{9} T^{23} + 43865 p^{10} T^{24} + 2438 p^{11} T^{25} + 5 p^{13} T^{26} + 13 p^{13} T^{27} + p^{14} T^{28} \)
61 \( 1 - 2 T + 531 T^{2} - 666 T^{3} + 138337 T^{4} - 79100 T^{5} + 23661761 T^{6} - 525585 T^{7} + 2997903312 T^{8} + 1049709712 T^{9} + 299443738513 T^{10} + 162692252758 T^{11} + 24337760215294 T^{12} + 14417559647767 T^{13} + 1631695419459942 T^{14} + 14417559647767 p T^{15} + 24337760215294 p^{2} T^{16} + 162692252758 p^{3} T^{17} + 299443738513 p^{4} T^{18} + 1049709712 p^{5} T^{19} + 2997903312 p^{6} T^{20} - 525585 p^{7} T^{21} + 23661761 p^{8} T^{22} - 79100 p^{9} T^{23} + 138337 p^{10} T^{24} - 666 p^{11} T^{25} + 531 p^{12} T^{26} - 2 p^{13} T^{27} + p^{14} T^{28} \)
67 \( 1 + 16 T + 519 T^{2} + 6672 T^{3} + 124668 T^{4} + 1350481 T^{5} + 19154411 T^{6} + 184611695 T^{7} + 2223612699 T^{8} + 19913080426 T^{9} + 215047481933 T^{10} + 1818762266026 T^{11} + 17963409855248 T^{12} + 142652630232610 T^{13} + 1296849511237522 T^{14} + 142652630232610 p T^{15} + 17963409855248 p^{2} T^{16} + 1818762266026 p^{3} T^{17} + 215047481933 p^{4} T^{18} + 19913080426 p^{5} T^{19} + 2223612699 p^{6} T^{20} + 184611695 p^{7} T^{21} + 19154411 p^{8} T^{22} + 1350481 p^{9} T^{23} + 124668 p^{10} T^{24} + 6672 p^{11} T^{25} + 519 p^{12} T^{26} + 16 p^{13} T^{27} + p^{14} T^{28} \)
71 \( 1 + 29 T + 705 T^{2} + 12086 T^{3} + 191896 T^{4} + 2638398 T^{5} + 34300098 T^{6} + 408688847 T^{7} + 4632044190 T^{8} + 49512503143 T^{9} + 506040747423 T^{10} + 4940191614446 T^{11} + 46145162701177 T^{12} + 414037169596841 T^{13} + 3556392873560764 T^{14} + 414037169596841 p T^{15} + 46145162701177 p^{2} T^{16} + 4940191614446 p^{3} T^{17} + 506040747423 p^{4} T^{18} + 49512503143 p^{5} T^{19} + 4632044190 p^{6} T^{20} + 408688847 p^{7} T^{21} + 34300098 p^{8} T^{22} + 2638398 p^{9} T^{23} + 191896 p^{10} T^{24} + 12086 p^{11} T^{25} + 705 p^{12} T^{26} + 29 p^{13} T^{27} + p^{14} T^{28} \)
73 \( 1 + 594 T^{2} - 544 T^{3} + 163050 T^{4} - 301055 T^{5} + 27556752 T^{6} - 79492103 T^{7} + 3229566043 T^{8} - 13470215088 T^{9} + 284293736642 T^{10} - 22602093480 p T^{11} + 20797805686594 T^{12} - 154352743930394 T^{13} + 1472400999295624 T^{14} - 154352743930394 p T^{15} + 20797805686594 p^{2} T^{16} - 22602093480 p^{4} T^{17} + 284293736642 p^{4} T^{18} - 13470215088 p^{5} T^{19} + 3229566043 p^{6} T^{20} - 79492103 p^{7} T^{21} + 27556752 p^{8} T^{22} - 301055 p^{9} T^{23} + 163050 p^{10} T^{24} - 544 p^{11} T^{25} + 594 p^{12} T^{26} + p^{14} T^{28} \)
79 \( 1 + 21 T + 775 T^{2} + 13429 T^{3} + 287452 T^{4} + 4258277 T^{5} + 68796789 T^{6} + 891779172 T^{7} + 11978875046 T^{8} + 138054011422 T^{9} + 1613060027341 T^{10} + 16698475768546 T^{11} + 173602203340189 T^{12} + 1622757633749715 T^{13} + 15178391862412142 T^{14} + 1622757633749715 p T^{15} + 173602203340189 p^{2} T^{16} + 16698475768546 p^{3} T^{17} + 1613060027341 p^{4} T^{18} + 138054011422 p^{5} T^{19} + 11978875046 p^{6} T^{20} + 891779172 p^{7} T^{21} + 68796789 p^{8} T^{22} + 4258277 p^{9} T^{23} + 287452 p^{10} T^{24} + 13429 p^{11} T^{25} + 775 p^{12} T^{26} + 21 p^{13} T^{27} + p^{14} T^{28} \)
83 \( 1 + 40 T + 1489 T^{2} + 36375 T^{3} + 815996 T^{4} + 14737335 T^{5} + 247530180 T^{6} + 3587969428 T^{7} + 49040298776 T^{8} + 598999223560 T^{9} + 6997481642155 T^{10} + 74730470575341 T^{11} + 774154667115923 T^{12} + 89730801745707 p T^{13} + 70193975652690912 T^{14} + 89730801745707 p^{2} T^{15} + 774154667115923 p^{2} T^{16} + 74730470575341 p^{3} T^{17} + 6997481642155 p^{4} T^{18} + 598999223560 p^{5} T^{19} + 49040298776 p^{6} T^{20} + 3587969428 p^{7} T^{21} + 247530180 p^{8} T^{22} + 14737335 p^{9} T^{23} + 815996 p^{10} T^{24} + 36375 p^{11} T^{25} + 1489 p^{12} T^{26} + 40 p^{13} T^{27} + p^{14} T^{28} \)
89 \( 1 + 48 T + 1746 T^{2} + 47342 T^{3} + 1100728 T^{4} + 22059597 T^{5} + 396813094 T^{6} + 6440871499 T^{7} + 95971599569 T^{8} + 1317084657840 T^{9} + 16797056961134 T^{10} + 199376003767338 T^{11} + 2213791916248950 T^{12} + 22996361109544056 T^{13} + 224121753113991348 T^{14} + 22996361109544056 p T^{15} + 2213791916248950 p^{2} T^{16} + 199376003767338 p^{3} T^{17} + 16797056961134 p^{4} T^{18} + 1317084657840 p^{5} T^{19} + 95971599569 p^{6} T^{20} + 6440871499 p^{7} T^{21} + 396813094 p^{8} T^{22} + 22059597 p^{9} T^{23} + 1100728 p^{10} T^{24} + 47342 p^{11} T^{25} + 1746 p^{12} T^{26} + 48 p^{13} T^{27} + p^{14} T^{28} \)
97 \( 1 - 18 T + 601 T^{2} - 9834 T^{3} + 207402 T^{4} - 2942663 T^{5} + 49182557 T^{6} - 631164431 T^{7} + 9013761651 T^{8} - 105528788834 T^{9} + 1343957536229 T^{10} - 14495773419210 T^{11} + 167194987366978 T^{12} - 1664560628547234 T^{13} + 17619370405805322 T^{14} - 1664560628547234 p T^{15} + 167194987366978 p^{2} T^{16} - 14495773419210 p^{3} T^{17} + 1343957536229 p^{4} T^{18} - 105528788834 p^{5} T^{19} + 9013761651 p^{6} T^{20} - 631164431 p^{7} T^{21} + 49182557 p^{8} T^{22} - 2942663 p^{9} T^{23} + 207402 p^{10} T^{24} - 9834 p^{11} T^{25} + 601 p^{12} T^{26} - 18 p^{13} T^{27} + p^{14} T^{28} \)
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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

−2.32986902366311676861853963242, −2.32696280735106909493050351957, −2.29023514978175452958804198951, −2.19917920367049794069490500641, −2.16524147815551817599955623821, −2.13612695106904058515930715364, −2.07270200427862947148767780617, −2.06902688469757288171233503230, −1.96983854029079652288219520623, −1.96597244260061186175447954534, −1.79041168928335264322328387509, −1.73449971740347984951326747604, −1.54447951269332685682479836098, −1.48464769958654406166094910247, −1.47944960258153677638518245169, −1.40145604728960189542194688654, −1.34642218403042467255483042642, −1.31747417707460182452185714681, −1.27756694770733209873428175938, −1.27542452568797168325844931754, −1.26544681117673891306055167251, −1.06282849101535722597941284503, −1.04879240734906847598177560747, −0.989210729909969253617787556756, −0.897430604946334436060671102123, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0.897430604946334436060671102123, 0.989210729909969253617787556756, 1.04879240734906847598177560747, 1.06282849101535722597941284503, 1.26544681117673891306055167251, 1.27542452568797168325844931754, 1.27756694770733209873428175938, 1.31747417707460182452185714681, 1.34642218403042467255483042642, 1.40145604728960189542194688654, 1.47944960258153677638518245169, 1.48464769958654406166094910247, 1.54447951269332685682479836098, 1.73449971740347984951326747604, 1.79041168928335264322328387509, 1.96597244260061186175447954534, 1.96983854029079652288219520623, 2.06902688469757288171233503230, 2.07270200427862947148767780617, 2.13612695106904058515930715364, 2.16524147815551817599955623821, 2.19917920367049794069490500641, 2.29023514978175452958804198951, 2.32696280735106909493050351957, 2.32986902366311676861853963242

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

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