Dirichlet series
| L(s) = 1 | − 16·5-s + 2·7-s − 9-s + 136·25-s − 2·27-s − 32·35-s + 12·37-s − 32·41-s + 32·43-s + 16·45-s − 4·47-s + 24·59-s − 2·63-s − 4·79-s − 81-s − 20·83-s + 24·89-s − 40·101-s + 36·109-s + 98·121-s − 816·125-s + 127-s + 131-s + 32·135-s + 137-s + 139-s + 149-s + ⋯ |
| L(s) = 1 | − 7.15·5-s + 0.755·7-s − 1/3·9-s + 27.1·25-s − 0.384·27-s − 5.40·35-s + 1.97·37-s − 4.99·41-s + 4.87·43-s + 2.38·45-s − 0.583·47-s + 3.12·59-s − 0.251·63-s − 0.450·79-s − 1/9·81-s − 2.19·83-s + 2.54·89-s − 3.98·101-s + 3.44·109-s + 8.90·121-s − 72.9·125-s + 0.0887·127-s + 0.0873·131-s + 2.75·135-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{16} \cdot 5^{16} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{16} \cdot 5^{16} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(32\) |
| Conductor: | \(2^{48} \cdot 3^{16} \cdot 5^{16} \cdot 7^{16}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.67842\times 10^{13}\) |
| Root analytic conductor: | \(2.58987\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((32,\ 2^{48} \cdot 3^{16} \cdot 5^{16} \cdot 7^{16} ,\ ( \ : [1/2]^{16} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.2606110516\) |
| \(L(\frac12)\) | \(\approx\) | \(0.2606110516\) |
| \(L(\frac{3}{2})\) | 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 | \( 1 + T^{2} + 2 T^{3} + 2 T^{4} - 2 T^{5} - p^{2} T^{6} + 20 T^{7} - 70 T^{8} + 20 p T^{9} - p^{4} T^{10} - 2 p^{3} T^{11} + 2 p^{4} T^{12} + 2 p^{5} T^{13} + p^{6} T^{14} + p^{8} T^{16} \) | |
| 5 | \( ( 1 + T )^{16} \) | |
| 7 | \( 1 - 2 T + 4 T^{2} - 34 T^{3} + 20 T^{4} + 54 T^{5} - 4 p^{2} T^{6} + 1590 T^{7} - 4714 T^{8} + 1590 p T^{9} - 4 p^{4} T^{10} + 54 p^{3} T^{11} + 20 p^{4} T^{12} - 34 p^{5} T^{13} + 4 p^{6} T^{14} - 2 p^{7} T^{15} + p^{8} T^{16} \) | |
| good | 11 | \( 1 - 98 T^{2} + 4677 T^{4} - 145778 T^{6} + 3365990 T^{8} - 61993422 T^{10} + 956138939 T^{12} - 12739504414 T^{14} + 149122019666 T^{16} - 12739504414 p^{2} T^{18} + 956138939 p^{4} T^{20} - 61993422 p^{6} T^{22} + 3365990 p^{8} T^{24} - 145778 p^{10} T^{26} + 4677 p^{12} T^{28} - 98 p^{14} T^{30} + p^{16} T^{32} \) |
| 13 | \( 1 - 62 T^{2} + 2189 T^{4} - 56298 T^{6} + 1188526 T^{8} - 21616414 T^{10} + 350156979 T^{12} - 5148156634 T^{14} + 69670246018 T^{16} - 5148156634 p^{2} T^{18} + 350156979 p^{4} T^{20} - 21616414 p^{6} T^{22} + 1188526 p^{8} T^{24} - 56298 p^{10} T^{26} + 2189 p^{12} T^{28} - 62 p^{14} T^{30} + p^{16} T^{32} \) | |
| 17 | \( ( 1 + 57 T^{2} - 98 T^{3} + 1646 T^{4} - 4282 T^{5} + 41047 T^{6} - 81412 T^{7} + 839106 T^{8} - 81412 p T^{9} + 41047 p^{2} T^{10} - 4282 p^{3} T^{11} + 1646 p^{4} T^{12} - 98 p^{5} T^{13} + 57 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 19 | \( 1 - 192 T^{2} + 18200 T^{4} - 1136320 T^{6} + 52494236 T^{8} - 1907281216 T^{10} + 56456539944 T^{12} - 1389647467328 T^{14} + 28750954230662 T^{16} - 1389647467328 p^{2} T^{18} + 56456539944 p^{4} T^{20} - 1907281216 p^{6} T^{22} + 52494236 p^{8} T^{24} - 1136320 p^{10} T^{26} + 18200 p^{12} T^{28} - 192 p^{14} T^{30} + p^{16} T^{32} \) | |
| 23 | \( 1 - 176 T^{2} + 16600 T^{4} - 1097616 T^{6} + 56000348 T^{8} - 2314599984 T^{10} + 79690491368 T^{12} - 2321865829264 T^{14} + 57730745485894 T^{16} - 2321865829264 p^{2} T^{18} + 79690491368 p^{4} T^{20} - 2314599984 p^{6} T^{22} + 56000348 p^{8} T^{24} - 1097616 p^{10} T^{26} + 16600 p^{12} T^{28} - 176 p^{14} T^{30} + p^{16} T^{32} \) | |
| 29 | \( 1 - 222 T^{2} + 26333 T^{4} - 2171770 T^{6} + 137753198 T^{8} - 7082097518 T^{10} + 10483939775 p T^{12} - 11089366548570 T^{14} + 346812558599170 T^{16} - 11089366548570 p^{2} T^{18} + 10483939775 p^{5} T^{20} - 7082097518 p^{6} T^{22} + 137753198 p^{8} T^{24} - 2171770 p^{10} T^{26} + 26333 p^{12} T^{28} - 222 p^{14} T^{30} + p^{16} T^{32} \) | |
| 31 | \( 1 - 236 T^{2} + 29072 T^{4} - 2456356 T^{6} + 159264268 T^{8} - 8416386060 T^{10} + 375234642352 T^{12} - 14382750977508 T^{14} + 478150921280358 T^{16} - 14382750977508 p^{2} T^{18} + 375234642352 p^{4} T^{20} - 8416386060 p^{6} T^{22} + 159264268 p^{8} T^{24} - 2456356 p^{10} T^{26} + 29072 p^{12} T^{28} - 236 p^{14} T^{30} + p^{16} T^{32} \) | |
| 37 | \( ( 1 - 6 T + 84 T^{2} - 770 T^{3} + 7124 T^{4} - 50334 T^{5} + 405452 T^{6} - 2547786 T^{7} + 16833206 T^{8} - 2547786 p T^{9} + 405452 p^{2} T^{10} - 50334 p^{3} T^{11} + 7124 p^{4} T^{12} - 770 p^{5} T^{13} + 84 p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 41 | \( ( 1 + 16 T + 308 T^{2} + 3264 T^{3} + 37460 T^{4} + 303968 T^{5} + 2637964 T^{6} + 17639760 T^{7} + 127095446 T^{8} + 17639760 p T^{9} + 2637964 p^{2} T^{10} + 303968 p^{3} T^{11} + 37460 p^{4} T^{12} + 3264 p^{5} T^{13} + 308 p^{6} T^{14} + 16 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 43 | \( ( 1 - 16 T + 348 T^{2} - 4144 T^{3} + 51940 T^{4} - 484528 T^{5} + 4416132 T^{6} - 33077136 T^{7} + 236220694 T^{8} - 33077136 p T^{9} + 4416132 p^{2} T^{10} - 484528 p^{3} T^{11} + 51940 p^{4} T^{12} - 4144 p^{5} T^{13} + 348 p^{6} T^{14} - 16 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 47 | \( ( 1 + 2 T + 177 T^{2} + 154 T^{3} + 17046 T^{4} + 6450 T^{5} + 1195423 T^{6} + 419226 T^{7} + 64751346 T^{8} + 419226 p T^{9} + 1195423 p^{2} T^{10} + 6450 p^{3} T^{11} + 17046 p^{4} T^{12} + 154 p^{5} T^{13} + 177 p^{6} T^{14} + 2 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 53 | \( 1 - 336 T^{2} + 57368 T^{4} - 6570480 T^{6} + 562971676 T^{8} - 721257872 p T^{10} + 2167136874280 T^{12} - 110899294229232 T^{14} + 5697668480853766 T^{16} - 110899294229232 p^{2} T^{18} + 2167136874280 p^{4} T^{20} - 721257872 p^{7} T^{22} + 562971676 p^{8} T^{24} - 6570480 p^{10} T^{26} + 57368 p^{12} T^{28} - 336 p^{14} T^{30} + p^{16} T^{32} \) | |
| 59 | \( ( 1 - 12 T + 328 T^{2} - 3436 T^{3} + 51916 T^{4} - 473868 T^{5} + 5241336 T^{6} - 40917388 T^{7} + 367036742 T^{8} - 40917388 p T^{9} + 5241336 p^{2} T^{10} - 473868 p^{3} T^{11} + 51916 p^{4} T^{12} - 3436 p^{5} T^{13} + 328 p^{6} T^{14} - 12 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 61 | \( 1 - 532 T^{2} + 140064 T^{4} - 24493116 T^{6} + 3211893356 T^{8} - 337476552116 T^{10} + 29598825516320 T^{12} - 2223593298357948 T^{14} + 145164479770038246 T^{16} - 2223593298357948 p^{2} T^{18} + 29598825516320 p^{4} T^{20} - 337476552116 p^{6} T^{22} + 3211893356 p^{8} T^{24} - 24493116 p^{10} T^{26} + 140064 p^{12} T^{28} - 532 p^{14} T^{30} + p^{16} T^{32} \) | |
| 67 | \( ( 1 + 220 T^{2} - 192 T^{3} + 27428 T^{4} - 59904 T^{5} + 2352452 T^{6} - 7304256 T^{7} + 167339734 T^{8} - 7304256 p T^{9} + 2352452 p^{2} T^{10} - 59904 p^{3} T^{11} + 27428 p^{4} T^{12} - 192 p^{5} T^{13} + 220 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 71 | \( 1 - 556 T^{2} + 159568 T^{4} - 31261348 T^{6} + 4675032076 T^{8} - 566675658188 T^{10} + 57715565878768 T^{12} - 5048338414984420 T^{14} + 383759996279289574 T^{16} - 5048338414984420 p^{2} T^{18} + 57715565878768 p^{4} T^{20} - 566675658188 p^{6} T^{22} + 4675032076 p^{8} T^{24} - 31261348 p^{10} T^{26} + 159568 p^{12} T^{28} - 556 p^{14} T^{30} + p^{16} T^{32} \) | |
| 73 | \( 1 - 484 T^{2} + 118496 T^{4} - 19741196 T^{6} + 2519941548 T^{8} - 263032335940 T^{10} + 23581346703712 T^{12} - 1898951836350284 T^{14} + 142510808022168230 T^{16} - 1898951836350284 p^{2} T^{18} + 23581346703712 p^{4} T^{20} - 263032335940 p^{6} T^{22} + 2519941548 p^{8} T^{24} - 19741196 p^{10} T^{26} + 118496 p^{12} T^{28} - 484 p^{14} T^{30} + p^{16} T^{32} \) | |
| 79 | \( ( 1 + 2 T + 193 T^{2} - 982 T^{3} + 17462 T^{4} - 211918 T^{5} + 1793391 T^{6} - 19574326 T^{7} + 178160242 T^{8} - 19574326 p T^{9} + 1793391 p^{2} T^{10} - 211918 p^{3} T^{11} + 17462 p^{4} T^{12} - 982 p^{5} T^{13} + 193 p^{6} T^{14} + 2 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 83 | \( ( 1 + 10 T + 476 T^{2} + 4914 T^{3} + 112516 T^{4} + 1075234 T^{5} + 16779076 T^{6} + 138590858 T^{7} + 1685572758 T^{8} + 138590858 p T^{9} + 16779076 p^{2} T^{10} + 1075234 p^{3} T^{11} + 112516 p^{4} T^{12} + 4914 p^{5} T^{13} + 476 p^{6} T^{14} + 10 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 89 | \( ( 1 - 12 T + 480 T^{2} - 5108 T^{3} + 107900 T^{4} - 1056060 T^{5} + 15486496 T^{6} - 137556996 T^{7} + 1596396230 T^{8} - 137556996 p T^{9} + 15486496 p^{2} T^{10} - 1056060 p^{3} T^{11} + 107900 p^{4} T^{12} - 5108 p^{5} T^{13} + 480 p^{6} T^{14} - 12 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 97 | \( 1 - 686 T^{2} + 252221 T^{4} - 64043882 T^{6} + 12508641022 T^{8} - 1992015843950 T^{10} + 268648339369171 T^{12} - 31445973528756858 T^{14} + 3242273163197169378 T^{16} - 31445973528756858 p^{2} T^{18} + 268648339369171 p^{4} T^{20} - 1992015843950 p^{6} T^{22} + 12508641022 p^{8} T^{24} - 64043882 p^{10} T^{26} + 252221 p^{12} T^{28} - 686 p^{14} T^{30} + p^{16} T^{32} \) | |
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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
−2.68707936948050471281288720202, −2.68160227957977986441458489736, −2.66750718454268042433328053453, −2.40829690165591560128071970864, −2.39019901346728191423073886203, −2.38658102409015489042688663434, −2.35273297174532368519890566742, −2.02891325124552686417288108058, −1.98368019499029006343151863337, −1.90226210185634149410349710260, −1.81020181298446147648591396874, −1.79961419355137801256099355185, −1.60648478446009938114882649163, −1.53832270122617932133760352360, −1.46221662182827388756188409947, −1.24952910940782362175057982107, −1.14583788040039758560325152889, −0.938164448712390297720542419552, −0.77556119170702829497037050778, −0.72658477238687432999552042431, −0.71899627596063368056906150782, −0.58315902034624030749907114461, −0.46291499561039230991169190515, −0.38795809214759296185524800454, −0.06919659456630221999986211040, 0.06919659456630221999986211040, 0.38795809214759296185524800454, 0.46291499561039230991169190515, 0.58315902034624030749907114461, 0.71899627596063368056906150782, 0.72658477238687432999552042431, 0.77556119170702829497037050778, 0.938164448712390297720542419552, 1.14583788040039758560325152889, 1.24952910940782362175057982107, 1.46221662182827388756188409947, 1.53832270122617932133760352360, 1.60648478446009938114882649163, 1.79961419355137801256099355185, 1.81020181298446147648591396874, 1.90226210185634149410349710260, 1.98368019499029006343151863337, 2.02891325124552686417288108058, 2.35273297174532368519890566742, 2.38658102409015489042688663434, 2.39019901346728191423073886203, 2.40829690165591560128071970864, 2.66750718454268042433328053453, 2.68160227957977986441458489736, 2.68707936948050471281288720202
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.