Dirichlet series
| L(s) = 1 | + 2·2-s + 16·3-s + 4-s + 32·6-s + 4·7-s + 136·9-s + 16·12-s + 8·14-s + 2·16-s + 272·18-s + 24·19-s + 64·21-s − 8·25-s + 816·27-s + 4·28-s + 16·29-s − 8·31-s + 136·36-s + 24·37-s + 48·38-s + 128·42-s − 16·47-s + 32·48-s − 16·50-s − 32·53-s + 1.63e3·54-s + 384·57-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 9.23·3-s + 1/2·4-s + 13.0·6-s + 1.51·7-s + 45.3·9-s + 4.61·12-s + 2.13·14-s + 1/2·16-s + 64.1·18-s + 5.50·19-s + 13.9·21-s − 8/5·25-s + 157.·27-s + 0.755·28-s + 2.97·29-s − 1.43·31-s + 68/3·36-s + 3.94·37-s + 7.78·38-s + 19.7·42-s − 2.33·47-s + 4.61·48-s − 2.26·50-s − 4.39·53-s + 222.·54-s + 50.8·57-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{16} \cdot 5^{16} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{16} \cdot 5^{16} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(32\) |
| Conductor: | \(2^{32} \cdot 3^{16} \cdot 5^{16} \cdot 7^{16}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.56107\times 10^{8}\) |
| Root analytic conductor: | \(1.83131\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((32,\ 2^{32} \cdot 3^{16} \cdot 5^{16} \cdot 7^{16} ,\ ( \ : [1/2]^{16} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(4178.563363\) |
| \(L(\frac12)\) | \(\approx\) | \(4178.563363\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 - p T + 3 T^{2} - p^{2} T^{3} + 3 T^{4} + p T^{5} - 7 T^{6} + 3 p^{2} T^{7} - 7 p^{2} T^{8} + 3 p^{3} T^{9} - 7 p^{2} T^{10} + p^{4} T^{11} + 3 p^{4} T^{12} - p^{7} T^{13} + 3 p^{6} T^{14} - p^{8} T^{15} + p^{8} T^{16} \) |
| 3 | \( ( 1 - T )^{16} \) | |
| 5 | \( ( 1 + T^{2} )^{8} \) | |
| 7 | \( 1 - 4 T + 16 T^{2} - 68 T^{3} + 188 T^{4} - 468 T^{5} + 1328 T^{6} - 3284 T^{7} + 6854 T^{8} - 3284 p T^{9} + 1328 p^{2} T^{10} - 468 p^{3} T^{11} + 188 p^{4} T^{12} - 68 p^{5} T^{13} + 16 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} \) | |
| good | 11 | \( 1 - 68 T^{2} + 2180 T^{4} - 46812 T^{6} + 798068 T^{8} - 11533812 T^{10} + 145791676 T^{12} - 1699254444 T^{14} + 19011779286 T^{16} - 1699254444 p^{2} T^{18} + 145791676 p^{4} T^{20} - 11533812 p^{6} T^{22} + 798068 p^{8} T^{24} - 46812 p^{10} T^{26} + 2180 p^{12} T^{28} - 68 p^{14} T^{30} + p^{16} T^{32} \) |
| 13 | \( 1 - 100 T^{2} + 5348 T^{4} - 198140 T^{6} + 5637172 T^{8} - 129840148 T^{10} + 2495923420 T^{12} - 40770066124 T^{14} + 571339339990 T^{16} - 40770066124 p^{2} T^{18} + 2495923420 p^{4} T^{20} - 129840148 p^{6} T^{22} + 5637172 p^{8} T^{24} - 198140 p^{10} T^{26} + 5348 p^{12} T^{28} - 100 p^{14} T^{30} + p^{16} T^{32} \) | |
| 17 | \( 1 - 168 T^{2} + 13880 T^{4} - 751096 T^{6} + 29923612 T^{8} - 935973160 T^{10} + 23941747080 T^{12} - 514532623928 T^{14} + 9440913455942 T^{16} - 514532623928 p^{2} T^{18} + 23941747080 p^{4} T^{20} - 935973160 p^{6} T^{22} + 29923612 p^{8} T^{24} - 751096 p^{10} T^{26} + 13880 p^{12} T^{28} - 168 p^{14} T^{30} + p^{16} T^{32} \) | |
| 19 | \( ( 1 - 12 T + 158 T^{2} - 1308 T^{3} + 10440 T^{4} - 66012 T^{5} + 391538 T^{6} - 1971116 T^{7} + 9248974 T^{8} - 1971116 p T^{9} + 391538 p^{2} T^{10} - 66012 p^{3} T^{11} + 10440 p^{4} T^{12} - 1308 p^{5} T^{13} + 158 p^{6} T^{14} - 12 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 23 | \( 1 - 176 T^{2} + 15448 T^{4} - 913680 T^{6} + 41467228 T^{8} - 1550033328 T^{10} + 49589827176 T^{12} - 1383348399760 T^{14} + 33907824809542 T^{16} - 1383348399760 p^{2} T^{18} + 49589827176 p^{4} T^{20} - 1550033328 p^{6} T^{22} + 41467228 p^{8} T^{24} - 913680 p^{10} T^{26} + 15448 p^{12} T^{28} - 176 p^{14} T^{30} + p^{16} T^{32} \) | |
| 29 | \( ( 1 - 8 T + 100 T^{2} - 360 T^{3} + 3380 T^{4} - 3064 T^{5} + 62140 T^{6} + 238824 T^{7} + 906806 T^{8} + 238824 p T^{9} + 62140 p^{2} T^{10} - 3064 p^{3} T^{11} + 3380 p^{4} T^{12} - 360 p^{5} T^{13} + 100 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 31 | \( ( 1 + 4 T + 98 T^{2} + 516 T^{3} + 5624 T^{4} + 35252 T^{5} + 225022 T^{6} + 1588916 T^{7} + 7569102 T^{8} + 1588916 p T^{9} + 225022 p^{2} T^{10} + 35252 p^{3} T^{11} + 5624 p^{4} T^{12} + 516 p^{5} T^{13} + 98 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 37 | \( ( 1 - 12 T + 188 T^{2} - 1604 T^{3} + 16708 T^{4} - 119548 T^{5} + 988388 T^{6} - 6004372 T^{7} + 1131038 p T^{8} - 6004372 p T^{9} + 988388 p^{2} T^{10} - 119548 p^{3} T^{11} + 16708 p^{4} T^{12} - 1604 p^{5} T^{13} + 188 p^{6} T^{14} - 12 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 41 | \( 1 - 184 T^{2} + 20472 T^{4} - 1751720 T^{6} + 121699868 T^{8} - 7243988536 T^{10} + 380165048008 T^{12} - 17911615742824 T^{14} + 768666981378246 T^{16} - 17911615742824 p^{2} T^{18} + 380165048008 p^{4} T^{20} - 7243988536 p^{6} T^{22} + 121699868 p^{8} T^{24} - 1751720 p^{10} T^{26} + 20472 p^{12} T^{28} - 184 p^{14} T^{30} + p^{16} T^{32} \) | |
| 43 | \( 1 - 352 T^{2} + 65784 T^{4} - 8492960 T^{6} + 836964316 T^{8} - 66242040800 T^{10} + 4333734314568 T^{12} - 238312249642272 T^{14} + 11112687909569926 T^{16} - 238312249642272 p^{2} T^{18} + 4333734314568 p^{4} T^{20} - 66242040800 p^{6} T^{22} + 836964316 p^{8} T^{24} - 8492960 p^{10} T^{26} + 65784 p^{12} T^{28} - 352 p^{14} T^{30} + p^{16} T^{32} \) | |
| 47 | \( ( 1 + 8 T + 200 T^{2} + 1480 T^{3} + 21484 T^{4} + 142376 T^{5} + 1546616 T^{6} + 9131112 T^{7} + 83479334 T^{8} + 9131112 p T^{9} + 1546616 p^{2} T^{10} + 142376 p^{3} T^{11} + 21484 p^{4} T^{12} + 1480 p^{5} T^{13} + 200 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 53 | \( ( 1 + 16 T + 6 p T^{2} + 3232 T^{3} + 39656 T^{4} + 299616 T^{5} + 2879090 T^{6} + 18163120 T^{7} + 161612366 T^{8} + 18163120 p T^{9} + 2879090 p^{2} T^{10} + 299616 p^{3} T^{11} + 39656 p^{4} T^{12} + 3232 p^{5} T^{13} + 6 p^{7} T^{14} + 16 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 59 | \( ( 1 + 4 T + 216 T^{2} + 212 T^{3} + 21484 T^{4} - 35116 T^{5} + 1537192 T^{6} - 4969884 T^{7} + 95982982 T^{8} - 4969884 p T^{9} + 1537192 p^{2} T^{10} - 35116 p^{3} T^{11} + 21484 p^{4} T^{12} + 212 p^{5} T^{13} + 216 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 61 | \( 1 - 424 T^{2} + 85912 T^{4} - 11360056 T^{6} + 1147532060 T^{8} - 97928338472 T^{10} + 7427530642088 T^{12} - 510650158156408 T^{14} + 32321138943506822 T^{16} - 510650158156408 p^{2} T^{18} + 7427530642088 p^{4} T^{20} - 97928338472 p^{6} T^{22} + 1147532060 p^{8} T^{24} - 11360056 p^{10} T^{26} + 85912 p^{12} T^{28} - 424 p^{14} T^{30} + p^{16} T^{32} \) | |
| 67 | \( 1 - 568 T^{2} + 166008 T^{4} - 32876264 T^{6} + 4920317404 T^{8} - 589683946296 T^{10} + 58602724633032 T^{12} - 4934611468716584 T^{14} + 356328084895246342 T^{16} - 4934611468716584 p^{2} T^{18} + 58602724633032 p^{4} T^{20} - 589683946296 p^{6} T^{22} + 4920317404 p^{8} T^{24} - 32876264 p^{10} T^{26} + 166008 p^{12} T^{28} - 568 p^{14} T^{30} + p^{16} T^{32} \) | |
| 71 | \( 1 - 444 T^{2} + 111268 T^{4} - 19597764 T^{6} + 2684602292 T^{8} - 301990393612 T^{10} + 28967151924188 T^{12} - 2434802829480948 T^{14} + 182479018420426774 T^{16} - 2434802829480948 p^{2} T^{18} + 28967151924188 p^{4} T^{20} - 301990393612 p^{6} T^{22} + 2684602292 p^{8} T^{24} - 19597764 p^{10} T^{26} + 111268 p^{12} T^{28} - 444 p^{14} T^{30} + p^{16} T^{32} \) | |
| 73 | \( 1 - 708 T^{2} + 243620 T^{4} - 54480956 T^{6} + 8938972724 T^{8} - 1152660018548 T^{10} + 122264196408284 T^{12} - 11016340703744524 T^{14} + 860580405090458646 T^{16} - 11016340703744524 p^{2} T^{18} + 122264196408284 p^{4} T^{20} - 1152660018548 p^{6} T^{22} + 8938972724 p^{8} T^{24} - 54480956 p^{10} T^{26} + 243620 p^{12} T^{28} - 708 p^{14} T^{30} + p^{16} T^{32} \) | |
| 79 | \( 1 - 776 T^{2} + 298264 T^{4} - 75873816 T^{6} + 14350843100 T^{8} - 2143612302472 T^{10} + 261769978394792 T^{12} - 26666593413127256 T^{14} + 2290442673574914502 T^{16} - 26666593413127256 p^{2} T^{18} + 261769978394792 p^{4} T^{20} - 2143612302472 p^{6} T^{22} + 14350843100 p^{8} T^{24} - 75873816 p^{10} T^{26} + 298264 p^{12} T^{28} - 776 p^{14} T^{30} + p^{16} T^{32} \) | |
| 83 | \( ( 1 + 4 T + 196 T^{2} - 268 T^{3} + 25748 T^{4} - 39276 T^{5} + 3021596 T^{6} - 7672636 T^{7} + 246037366 T^{8} - 7672636 p T^{9} + 3021596 p^{2} T^{10} - 39276 p^{3} T^{11} + 25748 p^{4} T^{12} - 268 p^{5} T^{13} + 196 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 89 | \( 1 - 864 T^{2} + 376504 T^{4} - 109545120 T^{6} + 23755420700 T^{8} - 4063039933920 T^{10} + 6361453333320 p T^{12} - 65504477701979168 T^{14} + 6355980862575907782 T^{16} - 65504477701979168 p^{2} T^{18} + 6361453333320 p^{5} T^{20} - 4063039933920 p^{6} T^{22} + 23755420700 p^{8} T^{24} - 109545120 p^{10} T^{26} + 376504 p^{12} T^{28} - 864 p^{14} T^{30} + p^{16} T^{32} \) | |
| 97 | \( 1 - 708 T^{2} + 266468 T^{4} - 70482684 T^{6} + 14473556148 T^{8} - 2426815167348 T^{10} + 342304459764508 T^{12} - 41313766425365708 T^{14} + 4305828967872497814 T^{16} - 41313766425365708 p^{2} T^{18} + 342304459764508 p^{4} T^{20} - 2426815167348 p^{6} T^{22} + 14473556148 p^{8} T^{24} - 70482684 p^{10} T^{26} + 266468 p^{12} T^{28} - 708 p^{14} T^{30} + p^{16} T^{32} \) | |
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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
−3.04144090348233652058146539184, −3.03713559735885517570425403295, −2.97974027380995383080739855081, −2.90330909851266986174204021352, −2.75326929970657835638808138807, −2.74102243529585764849907997057, −2.69192348355070452755255785167, −2.59837412813749162669053243933, −2.59032065535142308579168852566, −2.58519639144915520094548214525, −2.09124470444325012590511219387, −2.02114946998871874654637148811, −1.99539422190057795124183104029, −1.97519490763799672645078310599, −1.95950835276835462514412939595, −1.78042606484901084746052171429, −1.76483654993503255035298470487, −1.65570993276324022323700532366, −1.42632073323977938329004352148, −1.34631604872994339705235336820, −1.33251450773711396630767582563, −1.16360975656584479367777353260, −0.911978179778824542182047749213, −0.883572526905719778457234256240, −0.75719836602886166847766754723, 0.75719836602886166847766754723, 0.883572526905719778457234256240, 0.911978179778824542182047749213, 1.16360975656584479367777353260, 1.33251450773711396630767582563, 1.34631604872994339705235336820, 1.42632073323977938329004352148, 1.65570993276324022323700532366, 1.76483654993503255035298470487, 1.78042606484901084746052171429, 1.95950835276835462514412939595, 1.97519490763799672645078310599, 1.99539422190057795124183104029, 2.02114946998871874654637148811, 2.09124470444325012590511219387, 2.58519639144915520094548214525, 2.59032065535142308579168852566, 2.59837412813749162669053243933, 2.69192348355070452755255785167, 2.74102243529585764849907997057, 2.75326929970657835638808138807, 2.90330909851266986174204021352, 2.97974027380995383080739855081, 3.03713559735885517570425403295, 3.04144090348233652058146539184
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.