Dirichlet series
| L(s) = 1 | − 2·5-s + 13·9-s + 14·11-s − 22·19-s + 8·25-s − 44·29-s + 18·31-s + 14·41-s − 26·45-s + 17·49-s − 28·55-s − 64·59-s + 34·61-s + 30·71-s + 4·79-s + 80·81-s − 92·89-s + 44·95-s + 182·99-s + 32·101-s − 34·109-s + 29·121-s − 10·125-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 13/3·9-s + 4.22·11-s − 5.04·19-s + 8/5·25-s − 8.17·29-s + 3.23·31-s + 2.18·41-s − 3.87·45-s + 17/7·49-s − 3.77·55-s − 8.33·59-s + 4.35·61-s + 3.56·71-s + 0.450·79-s + 80/9·81-s − 9.75·89-s + 4.51·95-s + 18.2·99-s + 3.18·101-s − 3.25·109-s + 2.63·121-s − 0.894·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 5^{16} \cdot 23^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 5^{16} \cdot 23^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(32\) |
| Conductor: | \(2^{48} \cdot 5^{16} \cdot 23^{16}\) |
| Sign: | $1$ |
| Analytic conductor: | \(7.19513\times 10^{13}\) |
| Root analytic conductor: | \(2.71039\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((32,\ 2^{48} \cdot 5^{16} \cdot 23^{16} ,\ ( \ : [1/2]^{16} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(10.54377248\) |
| \(L(\frac12)\) | \(\approx\) | \(10.54377248\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + 2 T - 4 T^{2} - 14 T^{3} + 12 T^{4} + 6 T^{5} - 36 p T^{6} + 14 p T^{7} + 1462 T^{8} + 14 p^{2} T^{9} - 36 p^{3} T^{10} + 6 p^{3} T^{11} + 12 p^{4} T^{12} - 14 p^{5} T^{13} - 4 p^{6} T^{14} + 2 p^{7} T^{15} + p^{8} T^{16} \) | |
| 23 | \( ( 1 + T^{2} )^{8} \) | |
| good | 3 | \( 1 - 13 T^{2} + 89 T^{4} - 481 T^{6} + 83 p^{3} T^{8} - 8996 T^{10} + 32614 T^{12} - 109682 T^{14} + 341590 T^{16} - 109682 p^{2} T^{18} + 32614 p^{4} T^{20} - 8996 p^{6} T^{22} + 83 p^{11} T^{24} - 481 p^{10} T^{26} + 89 p^{12} T^{28} - 13 p^{14} T^{30} + p^{16} T^{32} \) |
| 7 | \( 1 - 17 T^{2} + 191 T^{4} - 2747 T^{6} + 29850 T^{8} - 259737 T^{10} + 2420865 T^{12} - 19386107 T^{14} + 133408906 T^{16} - 19386107 p^{2} T^{18} + 2420865 p^{4} T^{20} - 259737 p^{6} T^{22} + 29850 p^{8} T^{24} - 2747 p^{10} T^{26} + 191 p^{12} T^{28} - 17 p^{14} T^{30} + p^{16} T^{32} \) | |
| 11 | \( ( 1 - 7 T + 59 T^{2} - 291 T^{3} + 1636 T^{4} - 6511 T^{5} + 29057 T^{6} - 8961 p T^{7} + 33778 p T^{8} - 8961 p^{2} T^{9} + 29057 p^{2} T^{10} - 6511 p^{3} T^{11} + 1636 p^{4} T^{12} - 291 p^{5} T^{13} + 59 p^{6} T^{14} - 7 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 13 | \( 1 - 73 T^{2} + 3165 T^{4} - 100613 T^{6} + 2567065 T^{8} - 54745556 T^{10} + 1000257038 T^{12} - 15870909394 T^{14} + 220359338222 T^{16} - 15870909394 p^{2} T^{18} + 1000257038 p^{4} T^{20} - 54745556 p^{6} T^{22} + 2567065 p^{8} T^{24} - 100613 p^{10} T^{26} + 3165 p^{12} T^{28} - 73 p^{14} T^{30} + p^{16} T^{32} \) | |
| 17 | \( 1 - 125 T^{2} + 7651 T^{4} - 314263 T^{6} + 9986742 T^{8} - 265061677 T^{10} + 6098720493 T^{12} - 123634229311 T^{14} + 2226491176946 T^{16} - 123634229311 p^{2} T^{18} + 6098720493 p^{4} T^{20} - 265061677 p^{6} T^{22} + 9986742 p^{8} T^{24} - 314263 p^{10} T^{26} + 7651 p^{12} T^{28} - 125 p^{14} T^{30} + p^{16} T^{32} \) | |
| 19 | \( ( 1 + 11 T + 145 T^{2} + 1031 T^{3} + 8194 T^{4} + 45395 T^{5} + 277871 T^{6} + 1278303 T^{7} + 6382690 T^{8} + 1278303 p T^{9} + 277871 p^{2} T^{10} + 45395 p^{3} T^{11} + 8194 p^{4} T^{12} + 1031 p^{5} T^{13} + 145 p^{6} T^{14} + 11 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 29 | \( ( 1 + 22 T + 378 T^{2} + 4562 T^{3} + 46845 T^{4} + 396848 T^{5} + 2962818 T^{6} + 19103104 T^{7} + 110073276 T^{8} + 19103104 p T^{9} + 2962818 p^{2} T^{10} + 396848 p^{3} T^{11} + 46845 p^{4} T^{12} + 4562 p^{5} T^{13} + 378 p^{6} T^{14} + 22 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 31 | \( ( 1 - 9 T + 169 T^{2} - 1187 T^{3} + 13885 T^{4} - 83680 T^{5} + 741862 T^{6} - 3771778 T^{7} + 27243514 T^{8} - 3771778 p T^{9} + 741862 p^{2} T^{10} - 83680 p^{3} T^{11} + 13885 p^{4} T^{12} - 1187 p^{5} T^{13} + 169 p^{6} T^{14} - 9 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 37 | \( 1 - 364 T^{2} + 65788 T^{4} - 7871236 T^{6} + 699777076 T^{8} - 49092359564 T^{10} + 2813263592644 T^{12} - 134374519079876 T^{14} + 5407502454352982 T^{16} - 134374519079876 p^{2} T^{18} + 2813263592644 p^{4} T^{20} - 49092359564 p^{6} T^{22} + 699777076 p^{8} T^{24} - 7871236 p^{10} T^{26} + 65788 p^{12} T^{28} - 364 p^{14} T^{30} + p^{16} T^{32} \) | |
| 41 | \( ( 1 - 7 T + 165 T^{2} - 983 T^{3} + 15653 T^{4} - 81252 T^{5} + 992230 T^{6} - 4507386 T^{7} + 47027466 T^{8} - 4507386 p T^{9} + 992230 p^{2} T^{10} - 81252 p^{3} T^{11} + 15653 p^{4} T^{12} - 983 p^{5} T^{13} + 165 p^{6} T^{14} - 7 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 43 | \( 1 - 420 T^{2} + 89760 T^{4} - 12834188 T^{6} + 1366553900 T^{8} - 2662393644 p T^{10} + 7787383813216 T^{12} - 438089984906156 T^{14} + 20582097669052006 T^{16} - 438089984906156 p^{2} T^{18} + 7787383813216 p^{4} T^{20} - 2662393644 p^{7} T^{22} + 1366553900 p^{8} T^{24} - 12834188 p^{10} T^{26} + 89760 p^{12} T^{28} - 420 p^{14} T^{30} + p^{16} T^{32} \) | |
| 47 | \( 1 - 404 T^{2} + 79726 T^{4} - 10259396 T^{6} + 973543769 T^{8} - 73373660880 T^{10} + 4634838659686 T^{12} - 255536712678296 T^{14} + 12627164281272996 T^{16} - 255536712678296 p^{2} T^{18} + 4634838659686 p^{4} T^{20} - 73373660880 p^{6} T^{22} + 973543769 p^{8} T^{24} - 10259396 p^{10} T^{26} + 79726 p^{12} T^{28} - 404 p^{14} T^{30} + p^{16} T^{32} \) | |
| 53 | \( 1 - 420 T^{2} + 91460 T^{4} - 13699516 T^{6} + 1574228212 T^{8} - 146672044564 T^{10} + 215715660172 p T^{12} - 759048420942924 T^{14} + 43324410199486422 T^{16} - 759048420942924 p^{2} T^{18} + 215715660172 p^{5} T^{20} - 146672044564 p^{6} T^{22} + 1574228212 p^{8} T^{24} - 13699516 p^{10} T^{26} + 91460 p^{12} T^{28} - 420 p^{14} T^{30} + p^{16} T^{32} \) | |
| 59 | \( ( 1 + 32 T + 848 T^{2} + 15088 T^{3} + 233036 T^{4} + 2882448 T^{5} + 31690672 T^{6} + 293593088 T^{7} + 2441220422 T^{8} + 293593088 p T^{9} + 31690672 p^{2} T^{10} + 2882448 p^{3} T^{11} + 233036 p^{4} T^{12} + 15088 p^{5} T^{13} + 848 p^{6} T^{14} + 32 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 61 | \( ( 1 - 17 T + 537 T^{2} - 7051 T^{3} + 121722 T^{4} - 1266233 T^{5} + 15347679 T^{6} - 127547203 T^{7} + 1182647826 T^{8} - 127547203 p T^{9} + 15347679 p^{2} T^{10} - 1266233 p^{3} T^{11} + 121722 p^{4} T^{12} - 7051 p^{5} T^{13} + 537 p^{6} T^{14} - 17 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 67 | \( 1 - 708 T^{2} + 247484 T^{4} - 56662508 T^{6} + 9520717556 T^{8} - 1247190018884 T^{10} + 132049232892804 T^{12} - 11550160358877900 T^{14} + 844693675874325590 T^{16} - 11550160358877900 p^{2} T^{18} + 132049232892804 p^{4} T^{20} - 1247190018884 p^{6} T^{22} + 9520717556 p^{8} T^{24} - 56662508 p^{10} T^{26} + 247484 p^{12} T^{28} - 708 p^{14} T^{30} + p^{16} T^{32} \) | |
| 71 | \( ( 1 - 15 T + 315 T^{2} - 3473 T^{3} + 44217 T^{4} - 416512 T^{5} + 4119286 T^{6} - 34541386 T^{7} + 304903386 T^{8} - 34541386 p T^{9} + 4119286 p^{2} T^{10} - 416512 p^{3} T^{11} + 44217 p^{4} T^{12} - 3473 p^{5} T^{13} + 315 p^{6} T^{14} - 15 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 73 | \( 1 - 616 T^{2} + 179078 T^{4} - 32131496 T^{6} + 3885288265 T^{8} - 320202312208 T^{10} + 16693459592630 T^{12} - 391928743859232 T^{14} - 1221407121902412 T^{16} - 391928743859232 p^{2} T^{18} + 16693459592630 p^{4} T^{20} - 320202312208 p^{6} T^{22} + 3885288265 p^{8} T^{24} - 32131496 p^{10} T^{26} + 179078 p^{12} T^{28} - 616 p^{14} T^{30} + p^{16} T^{32} \) | |
| 79 | \( ( 1 - 2 T + 426 T^{2} - 122 T^{3} + 1028 p T^{4} + 114566 T^{5} + 9610486 T^{6} + 24091102 T^{7} + 846932422 T^{8} + 24091102 p T^{9} + 9610486 p^{2} T^{10} + 114566 p^{3} T^{11} + 1028 p^{5} T^{12} - 122 p^{5} T^{13} + 426 p^{6} T^{14} - 2 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 83 | \( 1 - 736 T^{2} + 267912 T^{4} - 63912352 T^{6} + 11230486268 T^{8} - 1555839029728 T^{10} + 178498322721272 T^{12} - 17644461144399776 T^{14} + 1546669067560589894 T^{16} - 17644461144399776 p^{2} T^{18} + 178498322721272 p^{4} T^{20} - 1555839029728 p^{6} T^{22} + 11230486268 p^{8} T^{24} - 63912352 p^{10} T^{26} + 267912 p^{12} T^{28} - 736 p^{14} T^{30} + p^{16} T^{32} \) | |
| 89 | \( ( 1 + 46 T + 1422 T^{2} + 31258 T^{3} + 564580 T^{4} + 8459430 T^{5} + 110456418 T^{6} + 1252191346 T^{7} + 12609417558 T^{8} + 1252191346 p T^{9} + 110456418 p^{2} T^{10} + 8459430 p^{3} T^{11} + 564580 p^{4} T^{12} + 31258 p^{5} T^{13} + 1422 p^{6} T^{14} + 46 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 97 | \( 1 - 1297 T^{2} + 800299 T^{4} - 312681035 T^{6} + 86840696194 T^{8} - 18236744855257 T^{10} + 3004767644563765 T^{12} - 397263968059954187 T^{14} + 42668832188128350650 T^{16} - 397263968059954187 p^{2} T^{18} + 3004767644563765 p^{4} T^{20} - 18236744855257 p^{6} T^{22} + 86840696194 p^{8} T^{24} - 312681035 p^{10} T^{26} + 800299 p^{12} T^{28} - 1297 p^{14} T^{30} + p^{16} T^{32} \) | |
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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
−2.55540973153079014254296748739, −2.49859164138832904845787712033, −2.48810551028589288751792343147, −2.34226709096507556304382769051, −2.23657521686644384362051442350, −2.15723246134015602230845822952, −2.10804595147466163768345107327, −2.07164642597222462659722304047, −1.92321355145543647751025490044, −1.85775040993853610130256576517, −1.73062528074813721344556672454, −1.57501676325003354652614005464, −1.55841226145620979598485741212, −1.51508009205966393609671318970, −1.48702979283524138756525661691, −1.36340740078396470469070592024, −1.36029963424695064067158615917, −1.27490613306044140183138081477, −1.03944065497005211650234093339, −0.973388223385464168789223950921, −0.70936609504300154655545125035, −0.60553210691568671448900935269, −0.51141566041035652768769298616, −0.30376559036992195216469201814, −0.17375867224342794536475916559, 0.17375867224342794536475916559, 0.30376559036992195216469201814, 0.51141566041035652768769298616, 0.60553210691568671448900935269, 0.70936609504300154655545125035, 0.973388223385464168789223950921, 1.03944065497005211650234093339, 1.27490613306044140183138081477, 1.36029963424695064067158615917, 1.36340740078396470469070592024, 1.48702979283524138756525661691, 1.51508009205966393609671318970, 1.55841226145620979598485741212, 1.57501676325003354652614005464, 1.73062528074813721344556672454, 1.85775040993853610130256576517, 1.92321355145543647751025490044, 2.07164642597222462659722304047, 2.10804595147466163768345107327, 2.15723246134015602230845822952, 2.23657521686644384362051442350, 2.34226709096507556304382769051, 2.48810551028589288751792343147, 2.49859164138832904845787712033, 2.55540973153079014254296748739
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.