Dirichlet series
| L(s) = 1 | − 8·7-s − 24·13-s + 16·19-s − 16·31-s + 16·37-s + 44·49-s − 24·61-s − 32·67-s + 56·73-s + 96·79-s + 6·81-s + 192·91-s + 16·97-s − 16·109-s + 60·121-s + 127-s + 131-s − 128·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 256·169-s + 173-s + ⋯ |
| L(s) = 1 | − 3.02·7-s − 6.65·13-s + 3.67·19-s − 2.87·31-s + 2.63·37-s + 44/7·49-s − 3.07·61-s − 3.90·67-s + 6.55·73-s + 10.8·79-s + 2/3·81-s + 20.1·91-s + 1.62·97-s − 1.53·109-s + 5.45·121-s + 0.0887·127-s + 0.0873·131-s − 11.0·133-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 19.6·169-s + 0.0760·173-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{64} \cdot 3^{16} \cdot 13^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{64} \cdot 3^{16} \cdot 13^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(32\) |
| Conductor: | \(2^{64} \cdot 3^{16} \cdot 13^{16}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.44340\times 10^{11}\) |
| Root analytic conductor: | \(2.23218\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((32,\ 2^{64} \cdot 3^{16} \cdot 13^{16} ,\ ( \ : [1/2]^{16} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.3095281855\) |
| \(L(\frac12)\) | \(\approx\) | \(0.3095281855\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 2 p T^{4} - 5 p^{2} T^{8} - 2 p^{5} T^{12} + p^{8} T^{16} \) | |
| 13 | \( ( 1 + 12 T + 88 T^{2} + 36 p T^{3} + 1875 T^{4} + 36 p^{2} T^{5} + 88 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| good | 5 | \( 1 + 4 p T^{4} - 614 T^{8} - 1232 p T^{12} + 499171 T^{16} - 1232 p^{5} T^{20} - 614 p^{8} T^{24} + 4 p^{13} T^{28} + p^{16} T^{32} \) |
| 7 | \( ( 1 + 4 T + 2 T^{2} - 40 T^{3} - 136 T^{4} - 4 p^{2} T^{5} + 264 T^{6} + 1860 T^{7} + 6283 T^{8} + 1860 p T^{9} + 264 p^{2} T^{10} - 4 p^{5} T^{11} - 136 p^{4} T^{12} - 40 p^{5} T^{13} + 2 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 11 | \( ( 1 - 30 T^{2} + 448 T^{4} - 4440 T^{6} + 43563 T^{8} - 4440 p^{2} T^{10} + 448 p^{4} T^{12} - 30 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 17 | \( 1 - 64 T^{2} + 1698 T^{4} - 27392 T^{6} + 484385 T^{8} - 11761728 T^{10} + 255581794 T^{12} - 4617900928 T^{14} + 78472961604 T^{16} - 4617900928 p^{2} T^{18} + 255581794 p^{4} T^{20} - 11761728 p^{6} T^{22} + 484385 p^{8} T^{24} - 27392 p^{10} T^{26} + 1698 p^{12} T^{28} - 64 p^{14} T^{30} + p^{16} T^{32} \) | |
| 19 | \( ( 1 - 8 T + 2 T^{2} + 272 T^{3} - 1360 T^{4} + 2408 T^{5} + 7368 T^{6} - 82152 T^{7} + 474187 T^{8} - 82152 p T^{9} + 7368 p^{2} T^{10} + 2408 p^{3} T^{11} - 1360 p^{4} T^{12} + 272 p^{5} T^{13} + 2 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 23 | \( 1 - 124 T^{2} + 8076 T^{4} - 359864 T^{6} + 12197750 T^{8} - 328926108 T^{10} + 7393235296 T^{12} - 151154175460 T^{14} + 3250403453475 T^{16} - 151154175460 p^{2} T^{18} + 7393235296 p^{4} T^{20} - 328926108 p^{6} T^{22} + 12197750 p^{8} T^{24} - 359864 p^{10} T^{26} + 8076 p^{12} T^{28} - 124 p^{14} T^{30} + p^{16} T^{32} \) | |
| 29 | \( 1 + 100 T^{2} + 150 p T^{4} + 115208 T^{6} + 2244953 T^{8} + 22210536 T^{10} - 955168034 T^{12} - 61617531188 T^{14} - 2051650936908 T^{16} - 61617531188 p^{2} T^{18} - 955168034 p^{4} T^{20} + 22210536 p^{6} T^{22} + 2244953 p^{8} T^{24} + 115208 p^{10} T^{26} + 150 p^{13} T^{28} + 100 p^{14} T^{30} + p^{16} T^{32} \) | |
| 31 | \( ( 1 + 8 T + 32 T^{2} - 8 T^{3} + 500 T^{4} + 11608 T^{5} + 76896 T^{6} + 198312 T^{7} + 81958 T^{8} + 198312 p T^{9} + 76896 p^{2} T^{10} + 11608 p^{3} T^{11} + 500 p^{4} T^{12} - 8 p^{5} T^{13} + 32 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 37 | \( ( 1 - 8 T + 32 T^{2} - 40 T^{3} + 317 T^{4} + 572 T^{5} - 13920 T^{6} + 285756 T^{7} - 2558072 T^{8} + 285756 p T^{9} - 13920 p^{2} T^{10} + 572 p^{3} T^{11} + 317 p^{4} T^{12} - 40 p^{5} T^{13} + 32 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 41 | \( 1 + 96 T^{2} + 9410 T^{4} + 608448 T^{6} + 38612545 T^{8} + 2063684352 T^{10} + 105547753922 T^{12} + 4798758331872 T^{14} + 207037559359684 T^{16} + 4798758331872 p^{2} T^{18} + 105547753922 p^{4} T^{20} + 2063684352 p^{6} T^{22} + 38612545 p^{8} T^{24} + 608448 p^{10} T^{26} + 9410 p^{12} T^{28} + 96 p^{14} T^{30} + p^{16} T^{32} \) | |
| 43 | \( ( 1 + 154 T^{2} + 14116 T^{4} + 908908 T^{6} + 44890315 T^{8} + 908908 p^{2} T^{10} + 14116 p^{4} T^{12} + 154 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 47 | \( 1 - 2872 T^{4} + 3290884 T^{8} + 7102487576 T^{12} - 16252261348538 T^{16} + 7102487576 p^{4} T^{20} + 3290884 p^{8} T^{24} - 2872 p^{12} T^{28} + p^{16} T^{32} \) | |
| 53 | \( ( 1 - 268 T^{2} + 36418 T^{4} - 3215200 T^{6} + 200754715 T^{8} - 3215200 p^{2} T^{10} + 36418 p^{4} T^{12} - 268 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 59 | \( 1 + 168 T^{2} + 20300 T^{4} + 1829856 T^{6} + 144087370 T^{8} + 10538264136 T^{10} + 722963466032 T^{12} + 47781811223496 T^{14} + 2903520365302771 T^{16} + 47781811223496 p^{2} T^{18} + 722963466032 p^{4} T^{20} + 10538264136 p^{6} T^{22} + 144087370 p^{8} T^{24} + 1829856 p^{10} T^{26} + 20300 p^{12} T^{28} + 168 p^{14} T^{30} + p^{16} T^{32} \) | |
| 61 | \( ( 1 + 12 T - 40 T^{2} - 552 T^{3} + 2509 T^{4} - 16788 T^{5} - 457240 T^{6} + 367560 T^{7} + 28033480 T^{8} + 367560 p T^{9} - 457240 p^{2} T^{10} - 16788 p^{3} T^{11} + 2509 p^{4} T^{12} - 552 p^{5} T^{13} - 40 p^{6} T^{14} + 12 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 67 | \( ( 1 + 16 T + 218 T^{2} + 2768 T^{3} + 26936 T^{4} + 288560 T^{5} + 2757624 T^{6} + 24343776 T^{7} + 218473003 T^{8} + 24343776 p T^{9} + 2757624 p^{2} T^{10} + 288560 p^{3} T^{11} + 26936 p^{4} T^{12} + 2768 p^{5} T^{13} + 218 p^{6} T^{14} + 16 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 71 | \( 1 + 12 T^{2} + 5012 T^{4} + 59568 T^{6} + 29972182 T^{8} + 85639092 T^{10} + 133919516768 T^{12} + 7825983434460 T^{14} + 889952663334499 T^{16} + 7825983434460 p^{2} T^{18} + 133919516768 p^{4} T^{20} + 85639092 p^{6} T^{22} + 29972182 p^{8} T^{24} + 59568 p^{10} T^{26} + 5012 p^{12} T^{28} + 12 p^{14} T^{30} + p^{16} T^{32} \) | |
| 73 | \( ( 1 - 28 T + 392 T^{2} - 4232 T^{3} + 37106 T^{4} - 243188 T^{5} + 1218624 T^{6} - 2086596 T^{7} - 23078141 T^{8} - 2086596 p T^{9} + 1218624 p^{2} T^{10} - 243188 p^{3} T^{11} + 37106 p^{4} T^{12} - 4232 p^{5} T^{13} + 392 p^{6} T^{14} - 28 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 79 | \( ( 1 - 24 T + 424 T^{2} - 5256 T^{3} + 54822 T^{4} - 5256 p T^{5} + 424 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} )^{4} \) | |
| 83 | \( 1 + 8744 T^{4} + 13736260 T^{8} - 185286399304 T^{12} - 1538291441783546 T^{16} - 185286399304 p^{4} T^{20} + 13736260 p^{8} T^{24} + 8744 p^{12} T^{28} + p^{16} T^{32} \) | |
| 89 | \( 1 + 264 T^{2} + 50372 T^{4} + 7164960 T^{6} + 840065098 T^{8} + 85008097032 T^{10} + 7787821049744 T^{12} + 683159083098984 T^{14} + 60065589678599059 T^{16} + 683159083098984 p^{2} T^{18} + 7787821049744 p^{4} T^{20} + 85008097032 p^{6} T^{22} + 840065098 p^{8} T^{24} + 7164960 p^{10} T^{26} + 50372 p^{12} T^{28} + 264 p^{14} T^{30} + p^{16} T^{32} \) | |
| 97 | \( ( 1 - 8 T + 296 T^{2} - 1384 T^{3} + 32750 T^{4} + 96992 T^{5} + 531408 T^{6} + 46914816 T^{7} - 133630205 T^{8} + 46914816 p T^{9} + 531408 p^{2} T^{10} + 96992 p^{3} T^{11} + 32750 p^{4} T^{12} - 1384 p^{5} T^{13} + 296 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
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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
−2.83637710787264024925222931084, −2.60520104271857448812968848213, −2.58454841647132047498691470690, −2.54754776833843621201789562614, −2.49198429777431768334851869650, −2.36163417082032982242917756800, −2.33483926608813321203485940911, −2.32879860020149250087190492593, −2.28714846226626619920987247336, −2.22801766979318118105000508340, −2.04046520252465164880483337398, −1.97209615854109791182815720512, −1.90968569861151687676764345781, −1.74922901957988657419277681386, −1.74630958802606834985780791345, −1.41648607087918997888036506502, −1.36549070182822092667886329154, −1.13443936615048352088504231463, −0.926853697972400186437514812271, −0.804558355952109496583874236877, −0.77909095998468494677921778817, −0.65602883869269338821104044489, −0.61391056320389551695068273760, −0.27790742184330532422488524846, −0.081847277570407172359488960339, 0.081847277570407172359488960339, 0.27790742184330532422488524846, 0.61391056320389551695068273760, 0.65602883869269338821104044489, 0.77909095998468494677921778817, 0.804558355952109496583874236877, 0.926853697972400186437514812271, 1.13443936615048352088504231463, 1.36549070182822092667886329154, 1.41648607087918997888036506502, 1.74630958802606834985780791345, 1.74922901957988657419277681386, 1.90968569861151687676764345781, 1.97209615854109791182815720512, 2.04046520252465164880483337398, 2.22801766979318118105000508340, 2.28714846226626619920987247336, 2.32879860020149250087190492593, 2.33483926608813321203485940911, 2.36163417082032982242917756800, 2.49198429777431768334851869650, 2.54754776833843621201789562614, 2.58454841647132047498691470690, 2.60520104271857448812968848213, 2.83637710787264024925222931084
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.