Properties

Label 32-370e16-1.1-c1e16-0-1
Degree $32$
Conductor $1.234\times 10^{41}$
Sign $1$
Analytic cond. $3.37024\times 10^{7}$
Root an. cond. $1.71885$
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  + 8·2-s − 3·3-s + 28·4-s − 24·6-s − 12·7-s + 48·8-s − 9-s − 6·11-s − 84·12-s − 6·13-s − 96·14-s + 6·16-s − 8·18-s − 3·19-s + 36·21-s − 48·22-s − 22·23-s − 144·24-s − 3·25-s − 48·26-s + 12·27-s − 336·28-s − 168·32-s + 18·33-s − 28·36-s + 16·37-s − 24·38-s + ⋯
L(s)  = 1  + 5.65·2-s − 1.73·3-s + 14·4-s − 9.79·6-s − 4.53·7-s + 16.9·8-s − 1/3·9-s − 1.80·11-s − 24.2·12-s − 1.66·13-s − 25.6·14-s + 3/2·16-s − 1.88·18-s − 0.688·19-s + 7.85·21-s − 10.2·22-s − 4.58·23-s − 29.3·24-s − 3/5·25-s − 9.41·26-s + 2.30·27-s − 63.4·28-s − 29.6·32-s + 3.13·33-s − 4.66·36-s + 2.63·37-s − 3.89·38-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{16} \cdot 37^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{16} \cdot 37^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(32\)
Conductor: \(2^{16} \cdot 5^{16} \cdot 37^{16}\)
Sign: $1$
Analytic conductor: \(3.37024\times 10^{7}\)
Root analytic conductor: \(1.71885\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((32,\ 2^{16} \cdot 5^{16} \cdot 37^{16} ,\ ( \ : [1/2]^{16} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.02978232988\)
\(L(\frac12)\) \(\approx\) \(0.02978232988\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 - T + T^{2} )^{8} \)
5 \( 1 + 3 T^{2} + 16 T^{3} - 11 T^{4} + 128 T^{5} + 142 T^{6} - 8 p T^{7} + 326 p T^{8} - 8 p^{2} T^{9} + 142 p^{2} T^{10} + 128 p^{3} T^{11} - 11 p^{4} T^{12} + 16 p^{5} T^{13} + 3 p^{6} T^{14} + p^{8} T^{16} \)
37 \( 1 - 16 T + 62 T^{2} + 274 T^{3} - 902 T^{4} - 12642 T^{5} + 1264 T^{6} + 921906 T^{7} - 7424265 T^{8} + 921906 p T^{9} + 1264 p^{2} T^{10} - 12642 p^{3} T^{11} - 902 p^{4} T^{12} + 274 p^{5} T^{13} + 62 p^{6} T^{14} - 16 p^{7} T^{15} + p^{8} T^{16} \)
good3 \( 1 + p T + 10 T^{2} + 7 p T^{3} + 31 T^{4} + 10 p T^{5} - 14 p T^{6} - 44 p T^{7} - 311 T^{8} - 67 p T^{9} + 488 T^{10} + 77 p^{3} T^{11} + 1330 p T^{12} + 103 p^{3} T^{13} - 1664 p T^{14} - 9569 p T^{15} - 59918 T^{16} - 9569 p^{2} T^{17} - 1664 p^{3} T^{18} + 103 p^{6} T^{19} + 1330 p^{5} T^{20} + 77 p^{8} T^{21} + 488 p^{6} T^{22} - 67 p^{8} T^{23} - 311 p^{8} T^{24} - 44 p^{10} T^{25} - 14 p^{11} T^{26} + 10 p^{12} T^{27} + 31 p^{12} T^{28} + 7 p^{14} T^{29} + 10 p^{14} T^{30} + p^{16} T^{31} + p^{16} T^{32} \)
7 \( 1 + 12 T + 2 p^{2} T^{2} + 600 T^{3} + 3096 T^{4} + 1986 p T^{5} + 55680 T^{6} + 202344 T^{7} + 672466 T^{8} + 2067840 T^{9} + 5905498 T^{10} + 15805362 T^{11} + 39958608 T^{12} + 13842030 p T^{13} + 231013506 T^{14} + 559380648 T^{15} + 1430753139 T^{16} + 559380648 p T^{17} + 231013506 p^{2} T^{18} + 13842030 p^{4} T^{19} + 39958608 p^{4} T^{20} + 15805362 p^{5} T^{21} + 5905498 p^{6} T^{22} + 2067840 p^{7} T^{23} + 672466 p^{8} T^{24} + 202344 p^{9} T^{25} + 55680 p^{10} T^{26} + 1986 p^{12} T^{27} + 3096 p^{12} T^{28} + 600 p^{13} T^{29} + 2 p^{16} T^{30} + 12 p^{15} T^{31} + p^{16} T^{32} \)
11 \( ( 1 + 3 T + 32 T^{2} + 3 p T^{3} + 490 T^{4} + 117 T^{5} + 6320 T^{6} - 3513 T^{7} + 64522 T^{8} - 3513 p T^{9} + 6320 p^{2} T^{10} + 117 p^{3} T^{11} + 490 p^{4} T^{12} + 3 p^{6} T^{13} + 32 p^{6} T^{14} + 3 p^{7} T^{15} + p^{8} T^{16} )^{2} \)
13 \( 1 + 6 T - 20 T^{2} + 20 T^{3} + 1151 T^{4} - 1654 T^{5} - 7024 T^{6} + 105492 T^{7} - 67135 T^{8} - 45230 p T^{9} + 551924 p T^{10} - 6996494 T^{11} - 32006354 T^{12} + 465371866 T^{13} - 568187972 T^{14} - 149876938 p T^{15} + 25325325778 T^{16} - 149876938 p^{2} T^{17} - 568187972 p^{2} T^{18} + 465371866 p^{3} T^{19} - 32006354 p^{4} T^{20} - 6996494 p^{5} T^{21} + 551924 p^{7} T^{22} - 45230 p^{8} T^{23} - 67135 p^{8} T^{24} + 105492 p^{9} T^{25} - 7024 p^{10} T^{26} - 1654 p^{11} T^{27} + 1151 p^{12} T^{28} + 20 p^{13} T^{29} - 20 p^{14} T^{30} + 6 p^{15} T^{31} + p^{16} T^{32} \)
17 \( 1 - 78 T^{2} - 12 p T^{3} + 3279 T^{4} + 14526 T^{5} - 71958 T^{6} - 550908 T^{7} + 580297 T^{8} + 12780882 T^{9} + 15260556 T^{10} - 192244386 T^{11} - 636642666 T^{12} + 1856646204 T^{13} + 12999911748 T^{14} - 502490754 p T^{15} - 13181871894 p T^{16} - 502490754 p^{2} T^{17} + 12999911748 p^{2} T^{18} + 1856646204 p^{3} T^{19} - 636642666 p^{4} T^{20} - 192244386 p^{5} T^{21} + 15260556 p^{6} T^{22} + 12780882 p^{7} T^{23} + 580297 p^{8} T^{24} - 550908 p^{9} T^{25} - 71958 p^{10} T^{26} + 14526 p^{11} T^{27} + 3279 p^{12} T^{28} - 12 p^{14} T^{29} - 78 p^{14} T^{30} + p^{16} T^{32} \)
19 \( 1 + 3 T + 61 T^{2} + 174 T^{3} + 1515 T^{4} + 93 p T^{5} + 13152 T^{6} - 65253 T^{7} - 177149 T^{8} - 1631784 T^{9} - 3303559 T^{10} - 218271 p T^{11} - 11699694 T^{12} + 86664717 T^{13} - 2011336563 T^{14} - 6695375232 T^{15} - 69017189232 T^{16} - 6695375232 p T^{17} - 2011336563 p^{2} T^{18} + 86664717 p^{3} T^{19} - 11699694 p^{4} T^{20} - 218271 p^{6} T^{21} - 3303559 p^{6} T^{22} - 1631784 p^{7} T^{23} - 177149 p^{8} T^{24} - 65253 p^{9} T^{25} + 13152 p^{10} T^{26} + 93 p^{12} T^{27} + 1515 p^{12} T^{28} + 174 p^{13} T^{29} + 61 p^{14} T^{30} + 3 p^{15} T^{31} + p^{16} T^{32} \)
23 \( ( 1 + 11 T + 128 T^{2} + 911 T^{3} + 6478 T^{4} + 38883 T^{5} + 225200 T^{6} + 1230055 T^{7} + 6051322 T^{8} + 1230055 p T^{9} + 225200 p^{2} T^{10} + 38883 p^{3} T^{11} + 6478 p^{4} T^{12} + 911 p^{5} T^{13} + 128 p^{6} T^{14} + 11 p^{7} T^{15} + p^{8} T^{16} )^{2} \)
29 \( 1 - 272 T^{2} + 36614 T^{4} - 3266476 T^{6} + 217791337 T^{8} - 11561581752 T^{10} + 506476572622 T^{12} - 18682924677900 T^{14} + 586411605736356 T^{16} - 18682924677900 p^{2} T^{18} + 506476572622 p^{4} T^{20} - 11561581752 p^{6} T^{22} + 217791337 p^{8} T^{24} - 3266476 p^{10} T^{26} + 36614 p^{12} T^{28} - 272 p^{14} T^{30} + p^{16} T^{32} \)
31 \( 1 - 375 T^{2} + 67893 T^{4} - 7906782 T^{6} + 664929132 T^{8} - 42908273244 T^{10} + 2201026606179 T^{12} - 91630932584847 T^{14} + 3130300401081590 T^{16} - 91630932584847 p^{2} T^{18} + 2201026606179 p^{4} T^{20} - 42908273244 p^{6} T^{22} + 664929132 p^{8} T^{24} - 7906782 p^{10} T^{26} + 67893 p^{12} T^{28} - 375 p^{14} T^{30} + p^{16} T^{32} \)
41 \( 1 - 7 T - 66 T^{2} - 221 T^{3} + 4518 T^{4} + 39871 T^{5} + 55294 T^{6} - 1501987 T^{7} - 15751275 T^{8} - 1049792 p T^{9} + 456677256 T^{10} + 3844650680 T^{11} + 13996654098 T^{12} - 69832479070 T^{13} - 808346463364 T^{14} + 323601268958 T^{15} + 13159594013156 T^{16} + 323601268958 p T^{17} - 808346463364 p^{2} T^{18} - 69832479070 p^{3} T^{19} + 13996654098 p^{4} T^{20} + 3844650680 p^{5} T^{21} + 456677256 p^{6} T^{22} - 1049792 p^{8} T^{23} - 15751275 p^{8} T^{24} - 1501987 p^{9} T^{25} + 55294 p^{10} T^{26} + 39871 p^{11} T^{27} + 4518 p^{12} T^{28} - 221 p^{13} T^{29} - 66 p^{14} T^{30} - 7 p^{15} T^{31} + p^{16} T^{32} \)
43 \( ( 1 + 11 T + 275 T^{2} + 2112 T^{3} + 31770 T^{4} + 194166 T^{5} + 2297629 T^{6} + 11906737 T^{7} + 117179090 T^{8} + 11906737 p T^{9} + 2297629 p^{2} T^{10} + 194166 p^{3} T^{11} + 31770 p^{4} T^{12} + 2112 p^{5} T^{13} + 275 p^{6} T^{14} + 11 p^{7} T^{15} + p^{8} T^{16} )^{2} \)
47 \( 1 - 385 T^{2} + 73210 T^{4} - 9245891 T^{6} + 883455844 T^{8} - 68838481361 T^{10} + 4565329257110 T^{12} - 262828743243691 T^{14} + 13213564960633878 T^{16} - 262828743243691 p^{2} T^{18} + 4565329257110 p^{4} T^{20} - 68838481361 p^{6} T^{22} + 883455844 p^{8} T^{24} - 9245891 p^{10} T^{26} + 73210 p^{12} T^{28} - 385 p^{14} T^{30} + p^{16} T^{32} \)
53 \( 1 - 3 T + 202 T^{2} - 597 T^{3} + 21187 T^{4} - 92490 T^{5} + 1253978 T^{6} - 9383418 T^{7} + 35037973 T^{8} - 12593973 p T^{9} - 348074944 T^{10} - 25850457075 T^{11} - 43358417578 T^{12} + 90396242733 T^{13} + 2109902381932 T^{14} + 78347485165647 T^{15} + 254945139522582 T^{16} + 78347485165647 p T^{17} + 2109902381932 p^{2} T^{18} + 90396242733 p^{3} T^{19} - 43358417578 p^{4} T^{20} - 25850457075 p^{5} T^{21} - 348074944 p^{6} T^{22} - 12593973 p^{8} T^{23} + 35037973 p^{8} T^{24} - 9383418 p^{9} T^{25} + 1253978 p^{10} T^{26} - 92490 p^{11} T^{27} + 21187 p^{12} T^{28} - 597 p^{13} T^{29} + 202 p^{14} T^{30} - 3 p^{15} T^{31} + p^{16} T^{32} \)
59 \( 1 - 15 T + 353 T^{2} - 4170 T^{3} + 58011 T^{4} - 522597 T^{5} + 5179566 T^{6} - 35960121 T^{7} + 262213231 T^{8} - 1391902122 T^{9} + 8415275263 T^{10} - 64302185301 T^{11} + 631394598906 T^{12} - 8645180571183 T^{13} + 85098631097223 T^{14} - 899523742922106 T^{15} + 6711635656764852 T^{16} - 899523742922106 p T^{17} + 85098631097223 p^{2} T^{18} - 8645180571183 p^{3} T^{19} + 631394598906 p^{4} T^{20} - 64302185301 p^{5} T^{21} + 8415275263 p^{6} T^{22} - 1391902122 p^{7} T^{23} + 262213231 p^{8} T^{24} - 35960121 p^{9} T^{25} + 5179566 p^{10} T^{26} - 522597 p^{11} T^{27} + 58011 p^{12} T^{28} - 4170 p^{13} T^{29} + 353 p^{14} T^{30} - 15 p^{15} T^{31} + p^{16} T^{32} \)
61 \( 1 - 12 T + 344 T^{2} - 3552 T^{3} + 54705 T^{4} - 491616 T^{5} + 5805956 T^{6} - 47479356 T^{7} + 509925713 T^{8} - 4002638592 T^{9} + 657536764 p T^{10} - 305449130112 T^{11} + 2803558979334 T^{12} - 20547894419112 T^{13} + 178906868507260 T^{14} - 1273829809567680 T^{15} + 10999532724872806 T^{16} - 1273829809567680 p T^{17} + 178906868507260 p^{2} T^{18} - 20547894419112 p^{3} T^{19} + 2803558979334 p^{4} T^{20} - 305449130112 p^{5} T^{21} + 657536764 p^{7} T^{22} - 4002638592 p^{7} T^{23} + 509925713 p^{8} T^{24} - 47479356 p^{9} T^{25} + 5805956 p^{10} T^{26} - 491616 p^{11} T^{27} + 54705 p^{12} T^{28} - 3552 p^{13} T^{29} + 344 p^{14} T^{30} - 12 p^{15} T^{31} + p^{16} T^{32} \)
67 \( 1 - 24 T + 484 T^{2} - 7008 T^{3} + 91460 T^{4} - 1036212 T^{5} + 10761864 T^{6} - 100498128 T^{7} + 840132906 T^{8} - 6141001536 T^{9} + 36092288364 T^{10} - 129503135916 T^{11} - 581769246320 T^{12} + 18321302475780 T^{13} - 246855146291380 T^{14} + 2575859195622768 T^{15} - 22616211153889357 T^{16} + 2575859195622768 p T^{17} - 246855146291380 p^{2} T^{18} + 18321302475780 p^{3} T^{19} - 581769246320 p^{4} T^{20} - 129503135916 p^{5} T^{21} + 36092288364 p^{6} T^{22} - 6141001536 p^{7} T^{23} + 840132906 p^{8} T^{24} - 100498128 p^{9} T^{25} + 10761864 p^{10} T^{26} - 1036212 p^{11} T^{27} + 91460 p^{12} T^{28} - 7008 p^{13} T^{29} + 484 p^{14} T^{30} - 24 p^{15} T^{31} + p^{16} T^{32} \)
71 \( 1 + 4 T - 300 T^{2} - 608 T^{3} + 50616 T^{4} + 120 p T^{5} - 5376024 T^{6} + 13609300 T^{7} + 371766658 T^{8} - 2680494096 T^{9} - 11122921492 T^{10} + 294691657624 T^{11} - 936862108288 T^{12} - 19834061419288 T^{13} + 179087837638140 T^{14} + 588749756030736 T^{15} - 15849592597843277 T^{16} + 588749756030736 p T^{17} + 179087837638140 p^{2} T^{18} - 19834061419288 p^{3} T^{19} - 936862108288 p^{4} T^{20} + 294691657624 p^{5} T^{21} - 11122921492 p^{6} T^{22} - 2680494096 p^{7} T^{23} + 371766658 p^{8} T^{24} + 13609300 p^{9} T^{25} - 5376024 p^{10} T^{26} + 120 p^{12} T^{27} + 50616 p^{12} T^{28} - 608 p^{13} T^{29} - 300 p^{14} T^{30} + 4 p^{15} T^{31} + p^{16} T^{32} \)
73 \( 1 - 520 T^{2} + 147408 T^{4} - 29456716 T^{6} + 4574760740 T^{8} - 580522082156 T^{10} + 61921919123088 T^{12} - 5643171350337152 T^{14} + 443177866810539382 T^{16} - 5643171350337152 p^{2} T^{18} + 61921919123088 p^{4} T^{20} - 580522082156 p^{6} T^{22} + 4574760740 p^{8} T^{24} - 29456716 p^{10} T^{26} + 147408 p^{12} T^{28} - 520 p^{14} T^{30} + p^{16} T^{32} \)
79 \( 1 + 428 T^{2} + 92092 T^{4} - 3444 T^{5} + 14348920 T^{6} - 1522680 T^{7} + 1869320218 T^{8} - 337986192 T^{9} + 209704238308 T^{10} - 52282569276 T^{11} + 20672365988528 T^{12} - 6342425533452 T^{13} + 1855147933628708 T^{14} - 620635034783328 T^{15} + 153189270589226835 T^{16} - 620635034783328 p T^{17} + 1855147933628708 p^{2} T^{18} - 6342425533452 p^{3} T^{19} + 20672365988528 p^{4} T^{20} - 52282569276 p^{5} T^{21} + 209704238308 p^{6} T^{22} - 337986192 p^{7} T^{23} + 1869320218 p^{8} T^{24} - 1522680 p^{9} T^{25} + 14348920 p^{10} T^{26} - 3444 p^{11} T^{27} + 92092 p^{12} T^{28} + 428 p^{14} T^{30} + p^{16} T^{32} \)
83 \( 1 - 6 T + 416 T^{2} - 2424 T^{3} + 88044 T^{4} - 609798 T^{5} + 13505904 T^{6} - 117246042 T^{7} + 1727480710 T^{8} - 17705306820 T^{9} + 198900837952 T^{10} - 2186513494746 T^{11} + 21149515848072 T^{12} - 229179984026142 T^{13} + 2052571851217104 T^{14} - 21114086156488716 T^{15} + 179724537353291415 T^{16} - 21114086156488716 p T^{17} + 2052571851217104 p^{2} T^{18} - 229179984026142 p^{3} T^{19} + 21149515848072 p^{4} T^{20} - 2186513494746 p^{5} T^{21} + 198900837952 p^{6} T^{22} - 17705306820 p^{7} T^{23} + 1727480710 p^{8} T^{24} - 117246042 p^{9} T^{25} + 13505904 p^{10} T^{26} - 609798 p^{11} T^{27} + 88044 p^{12} T^{28} - 2424 p^{13} T^{29} + 416 p^{14} T^{30} - 6 p^{15} T^{31} + p^{16} T^{32} \)
89 \( 1 - 9 T + 389 T^{2} - 3258 T^{3} + 69014 T^{4} - 526974 T^{5} + 7393077 T^{6} - 58179681 T^{7} + 650564997 T^{8} - 6634496976 T^{9} + 71318066208 T^{10} - 798885055992 T^{11} + 8672710581970 T^{12} - 78226953588990 T^{13} + 883787720311102 T^{14} - 6352317264744144 T^{15} + 79634251201787348 T^{16} - 6352317264744144 p T^{17} + 883787720311102 p^{2} T^{18} - 78226953588990 p^{3} T^{19} + 8672710581970 p^{4} T^{20} - 798885055992 p^{5} T^{21} + 71318066208 p^{6} T^{22} - 6634496976 p^{7} T^{23} + 650564997 p^{8} T^{24} - 58179681 p^{9} T^{25} + 7393077 p^{10} T^{26} - 526974 p^{11} T^{27} + 69014 p^{12} T^{28} - 3258 p^{13} T^{29} + 389 p^{14} T^{30} - 9 p^{15} T^{31} + p^{16} T^{32} \)
97 \( ( 1 - 34 T + 914 T^{2} - 16070 T^{3} + 241309 T^{4} - 2813736 T^{5} + 30235282 T^{6} - 281843400 T^{7} + 2819678604 T^{8} - 281843400 p T^{9} + 30235282 p^{2} T^{10} - 2813736 p^{3} T^{11} + 241309 p^{4} T^{12} - 16070 p^{5} T^{13} + 914 p^{6} T^{14} - 34 p^{7} T^{15} + p^{8} T^{16} )^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{32} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−3.32374666453571105584595711735, −3.27502055199344352323306096304, −2.99776668336749790693537820977, −2.97123310016518007810781102637, −2.95129898351160257170636171160, −2.89193064821544600395821119516, −2.75745533874750648033538194330, −2.58860408430699441186634220897, −2.56933877424371588213501252712, −2.53976605369694007338187521746, −2.39616603133974675002026766092, −2.36355127407092143965469883608, −2.22194744295710784921686144164, −2.21605536801492708899997198469, −2.07611926007148272229989514007, −2.03844474809102523591669482285, −1.75287909571347759162636645697, −1.72644212642618854903264326448, −1.25238045349816064749021237925, −1.14076848845333570410873036781, −0.852963131243367721629592293188, −0.817603354408658210194352680310, −0.20560568494274588677673347457, −0.17672656714296694781328097280, −0.092693275102365780925854816860, 0.092693275102365780925854816860, 0.17672656714296694781328097280, 0.20560568494274588677673347457, 0.817603354408658210194352680310, 0.852963131243367721629592293188, 1.14076848845333570410873036781, 1.25238045349816064749021237925, 1.72644212642618854903264326448, 1.75287909571347759162636645697, 2.03844474809102523591669482285, 2.07611926007148272229989514007, 2.21605536801492708899997198469, 2.22194744295710784921686144164, 2.36355127407092143965469883608, 2.39616603133974675002026766092, 2.53976605369694007338187521746, 2.56933877424371588213501252712, 2.58860408430699441186634220897, 2.75745533874750648033538194330, 2.89193064821544600395821119516, 2.95129898351160257170636171160, 2.97123310016518007810781102637, 2.99776668336749790693537820977, 3.27502055199344352323306096304, 3.32374666453571105584595711735

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

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