Dirichlet series
| L(s) = 1 | − 4·4-s + 4·11-s + 8·16-s − 4·29-s − 8·32-s − 20·37-s + 60·43-s − 16·44-s + 28·53-s − 16·64-s + 12·67-s − 72·79-s + 30·81-s − 12·107-s − 60·109-s + 16·113-s + 16·116-s + 8·121-s + 127-s + 48·128-s + 131-s + 137-s + 139-s + 80·148-s + 149-s + 151-s + 157-s + ⋯ |
| L(s) = 1 | − 2·4-s + 1.20·11-s + 2·16-s − 0.742·29-s − 1.41·32-s − 3.28·37-s + 9.14·43-s − 2.41·44-s + 3.84·53-s − 2·64-s + 1.46·67-s − 8.10·79-s + 10/3·81-s − 1.16·107-s − 5.74·109-s + 1.50·113-s + 1.48·116-s + 8/11·121-s + 0.0887·127-s + 4.24·128-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 6.57·148-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{80} \cdot 7^{40}\right)^{s/2} \, \Gamma_{\C}(s)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{80} \cdot 7^{40}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(40\) |
| Conductor: | \(2^{80} \cdot 7^{40}\) |
| Sign: | $1$ |
| Analytic conductor: | \(8.54797\times 10^{15}\) |
| Root analytic conductor: | \(2.50205\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((40,\ 2^{80} \cdot 7^{40} ,\ ( \ : [1/2]^{20} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(6.523960434\) |
| \(L(\frac12)\) | \(\approx\) | \(6.523960434\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 + p T^{2} + p T^{4} + p^{2} T^{5} + p^{2} T^{6} + p^{4} T^{8} + p^{5} T^{10} )^{2} \) |
| 7 | \( 1 \) | |
| good | 3 | \( 1 - 10 p T^{4} + 565 T^{8} - 2920 p T^{12} + 105514 T^{16} - 1025204 T^{20} + 105514 p^{4} T^{24} - 2920 p^{9} T^{28} + 565 p^{12} T^{32} - 10 p^{17} T^{36} + p^{20} T^{40} \) |
| 5 | \( 1 - 94 T^{4} + 4117 T^{8} - 131128 T^{12} + 3986026 T^{16} - 109987636 T^{20} + 3986026 p^{4} T^{24} - 131128 p^{8} T^{28} + 4117 p^{12} T^{32} - 94 p^{16} T^{36} + p^{20} T^{40} \) | |
| 11 | \( ( 1 - 2 T + 2 T^{2} - 6 T^{3} + 29 T^{4} - 24 T^{5} + 8 T^{6} + 1560 T^{7} - 18134 T^{8} + 21828 T^{9} - 16692 T^{10} + 21828 p T^{11} - 18134 p^{2} T^{12} + 1560 p^{3} T^{13} + 8 p^{4} T^{14} - 24 p^{5} T^{15} + 29 p^{6} T^{16} - 6 p^{7} T^{17} + 2 p^{8} T^{18} - 2 p^{9} T^{19} + p^{10} T^{20} )^{2} \) | |
| 13 | \( 1 - 510 T^{4} + 70453 T^{8} + 1978952 T^{12} - 31955094 T^{16} - 225295109620 T^{20} - 31955094 p^{4} T^{24} + 1978952 p^{8} T^{28} + 70453 p^{12} T^{32} - 510 p^{16} T^{36} + p^{20} T^{40} \) | |
| 17 | \( ( 1 + 58 T^{2} + 2237 T^{4} + 62552 T^{6} + 1425826 T^{8} + 26275932 T^{10} + 1425826 p^{2} T^{12} + 62552 p^{4} T^{14} + 2237 p^{6} T^{16} + 58 p^{8} T^{18} + p^{10} T^{20} )^{2} \) | |
| 19 | \( 1 - 1374 T^{4} + 1167221 T^{8} - 691865912 T^{12} + 334972765610 T^{16} - 131355341902388 T^{20} + 334972765610 p^{4} T^{24} - 691865912 p^{8} T^{28} + 1167221 p^{12} T^{32} - 1374 p^{16} T^{36} + p^{20} T^{40} \) | |
| 23 | \( ( 1 - 130 T^{2} + 8037 T^{4} - 322696 T^{6} + 9750842 T^{8} - 242809612 T^{10} + 9750842 p^{2} T^{12} - 322696 p^{4} T^{14} + 8037 p^{6} T^{16} - 130 p^{8} T^{18} + p^{10} T^{20} )^{2} \) | |
| 29 | \( ( 1 + 2 T + 2 T^{2} + 186 T^{3} + 581 T^{4} - 3720 T^{5} + 8696 T^{6} + 104856 T^{7} - 1195646 T^{8} - 5985588 T^{9} + 11790156 T^{10} - 5985588 p T^{11} - 1195646 p^{2} T^{12} + 104856 p^{3} T^{13} + 8696 p^{4} T^{14} - 3720 p^{5} T^{15} + 581 p^{6} T^{16} + 186 p^{7} T^{17} + 2 p^{8} T^{18} + 2 p^{9} T^{19} + p^{10} T^{20} )^{2} \) | |
| 31 | \( ( 1 + 54 T^{2} + 3357 T^{4} + 148200 T^{6} + 6003170 T^{8} + 187113284 T^{10} + 6003170 p^{2} T^{12} + 148200 p^{4} T^{14} + 3357 p^{6} T^{16} + 54 p^{8} T^{18} + p^{10} T^{20} )^{2} \) | |
| 37 | \( ( 1 + 10 T + 50 T^{2} + 10 p T^{3} + 2293 T^{4} + 8216 T^{5} + 35960 T^{6} + 197496 T^{7} - 138014 T^{8} - 5024644 T^{9} - 18977172 T^{10} - 5024644 p T^{11} - 138014 p^{2} T^{12} + 197496 p^{3} T^{13} + 35960 p^{4} T^{14} + 8216 p^{5} T^{15} + 2293 p^{6} T^{16} + 10 p^{8} T^{17} + 50 p^{8} T^{18} + 10 p^{9} T^{19} + p^{10} T^{20} )^{2} \) | |
| 41 | \( ( 1 - 162 T^{2} + 12269 T^{4} - 699192 T^{6} + 38412258 T^{8} - 1806374732 T^{10} + 38412258 p^{2} T^{12} - 699192 p^{4} T^{14} + 12269 p^{6} T^{16} - 162 p^{8} T^{18} + p^{10} T^{20} )^{2} \) | |
| 43 | \( ( 1 - 30 T + 450 T^{2} - 4874 T^{3} + 48157 T^{4} - 460984 T^{5} + 4036808 T^{6} - 740536 p T^{7} + 236888810 T^{8} - 1705241204 T^{9} + 11612848844 T^{10} - 1705241204 p T^{11} + 236888810 p^{2} T^{12} - 740536 p^{4} T^{13} + 4036808 p^{4} T^{14} - 460984 p^{5} T^{15} + 48157 p^{6} T^{16} - 4874 p^{7} T^{17} + 450 p^{8} T^{18} - 30 p^{9} T^{19} + p^{10} T^{20} )^{2} \) | |
| 47 | \( ( 1 + 214 T^{2} + 24701 T^{4} + 2083048 T^{6} + 136266594 T^{8} + 151153724 p T^{10} + 136266594 p^{2} T^{12} + 2083048 p^{4} T^{14} + 24701 p^{6} T^{16} + 214 p^{8} T^{18} + p^{10} T^{20} )^{2} \) | |
| 53 | \( ( 1 - 14 T + 98 T^{2} - 38 T^{3} - 3819 T^{4} - 8520 T^{5} + 494264 T^{6} - 5395816 T^{7} + 26807906 T^{8} - 9826516 T^{9} - 360673140 T^{10} - 9826516 p T^{11} + 26807906 p^{2} T^{12} - 5395816 p^{3} T^{13} + 494264 p^{4} T^{14} - 8520 p^{5} T^{15} - 3819 p^{6} T^{16} - 38 p^{7} T^{17} + 98 p^{8} T^{18} - 14 p^{9} T^{19} + p^{10} T^{20} )^{2} \) | |
| 59 | \( 1 - 4222 T^{4} + 35328725 T^{8} - 166476497848 T^{12} + 657120583021994 T^{16} - 2744188085716522484 T^{20} + 657120583021994 p^{4} T^{24} - 166476497848 p^{8} T^{28} + 35328725 p^{12} T^{32} - 4222 p^{16} T^{36} + p^{20} T^{40} \) | |
| 61 | \( 1 - 6206 T^{4} + 68327029 T^{8} - 325050414264 T^{12} + 1890122259799018 T^{16} - 6621011267052679540 T^{20} + 1890122259799018 p^{4} T^{24} - 325050414264 p^{8} T^{28} + 68327029 p^{12} T^{32} - 6206 p^{16} T^{36} + p^{20} T^{40} \) | |
| 67 | \( ( 1 - 6 T + 18 T^{2} - 674 T^{3} + 6413 T^{4} + 17784 T^{5} + 5000 T^{6} - 906936 T^{7} - 36633014 T^{8} + 348248972 T^{9} - 692754388 T^{10} + 348248972 p T^{11} - 36633014 p^{2} T^{12} - 906936 p^{3} T^{13} + 5000 p^{4} T^{14} + 17784 p^{5} T^{15} + 6413 p^{6} T^{16} - 674 p^{7} T^{17} + 18 p^{8} T^{18} - 6 p^{9} T^{19} + p^{10} T^{20} )^{2} \) | |
| 71 | \( ( 1 - 470 T^{2} + 110269 T^{4} - 16846024 T^{6} + 1844714482 T^{8} - 150749452100 T^{10} + 1844714482 p^{2} T^{12} - 16846024 p^{4} T^{14} + 110269 p^{6} T^{16} - 470 p^{8} T^{18} + p^{10} T^{20} )^{2} \) | |
| 73 | \( ( 1 - 354 T^{2} + 66813 T^{4} - 9017496 T^{6} + 928958930 T^{8} - 75536345228 T^{10} + 928958930 p^{2} T^{12} - 9017496 p^{4} T^{14} + 66813 p^{6} T^{16} - 354 p^{8} T^{18} + p^{10} T^{20} )^{2} \) | |
| 79 | \( ( 1 + 18 T + 403 T^{2} + 56 p T^{3} + 59042 T^{4} + 476204 T^{5} + 59042 p T^{6} + 56 p^{3} T^{7} + 403 p^{3} T^{8} + 18 p^{4} T^{9} + p^{5} T^{10} )^{4} \) | |
| 83 | \( 1 + 9762 T^{4} - 14543627 T^{8} - 610990879544 T^{12} - 966524108214486 T^{16} + 20156745060284242636 T^{20} - 966524108214486 p^{4} T^{24} - 610990879544 p^{8} T^{28} - 14543627 p^{12} T^{32} + 9762 p^{16} T^{36} + p^{20} T^{40} \) | |
| 89 | \( ( 1 - 610 T^{2} + 172893 T^{4} - 30661656 T^{6} + 3903499154 T^{8} - 387471505548 T^{10} + 3903499154 p^{2} T^{12} - 30661656 p^{4} T^{14} + 172893 p^{6} T^{16} - 610 p^{8} T^{18} + p^{10} T^{20} )^{2} \) | |
| 97 | \( ( 1 + 378 T^{2} + 84573 T^{4} + 13304280 T^{6} + 1672365986 T^{8} + 175388398556 T^{10} + 1672365986 p^{2} T^{12} + 13304280 p^{4} T^{14} + 84573 p^{6} T^{16} + 378 p^{8} T^{18} + p^{10} T^{20} )^{2} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{40} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.32101076904256038566779919856, −2.29610493869133446813675316574, −2.25946963351308330018885006562, −2.17448240063291097185504102190, −2.13955502934675167213349125450, −2.10522673347543807191091769306, −1.81905801747108427286370792736, −1.69690242491223955585895503530, −1.65972477136621967543372523463, −1.60830205585409279774305876618, −1.58115753581817366068750651973, −1.49001153242741790043509101458, −1.47081174161179226006223094099, −1.42123096648977231806250938723, −1.18207546765804631678212268213, −1.16033988814194788827582450778, −1.02233299851031460803206323509, −0.930777757255715092360971319477, −0.869749198345323022139443834058, −0.78384874442980361621038003551, −0.67295518714322336880446614723, −0.56267726919385852223124118415, −0.29411068080611782975319319055, −0.25714202177013397579714594817, −0.24463727827314911482778084183, 0.24463727827314911482778084183, 0.25714202177013397579714594817, 0.29411068080611782975319319055, 0.56267726919385852223124118415, 0.67295518714322336880446614723, 0.78384874442980361621038003551, 0.869749198345323022139443834058, 0.930777757255715092360971319477, 1.02233299851031460803206323509, 1.16033988814194788827582450778, 1.18207546765804631678212268213, 1.42123096648977231806250938723, 1.47081174161179226006223094099, 1.49001153242741790043509101458, 1.58115753581817366068750651973, 1.60830205585409279774305876618, 1.65972477136621967543372523463, 1.69690242491223955585895503530, 1.81905801747108427286370792736, 2.10522673347543807191091769306, 2.13955502934675167213349125450, 2.17448240063291097185504102190, 2.25946963351308330018885006562, 2.29610493869133446813675316574, 2.32101076904256038566779919856
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.