Dirichlet series
| L(s) = 1 | − 1.31e5·2-s + 3.64e7·3-s + 1.71e10·4-s + 3.89e11·5-s − 4.78e12·6-s − 1.29e14·7-s − 2.25e15·8-s − 4.86e16·9-s − 5.09e16·10-s + 4.69e17·11-s + 6.26e17·12-s + 2.85e19·13-s + 1.69e19·14-s + 1.41e19·15-s + 2.95e20·16-s + 5.39e21·17-s + 6.38e21·18-s + 3.41e22·19-s + 6.68e21·20-s − 4.73e21·21-s − 6.15e22·22-s − 1.70e23·23-s − 8.21e22·24-s − 2.75e24·25-s − 3.74e24·26-s − 3.60e24·27-s − 2.22e24·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.163·3-s + 1/2·4-s + 0.228·5-s − 0.115·6-s − 0.210·7-s − 0.353·8-s − 0.973·9-s − 0.161·10-s + 0.280·11-s + 0.0815·12-s + 0.915·13-s + 0.148·14-s + 0.0372·15-s + 1/4·16-s + 1.58·17-s + 0.688·18-s + 1.43·19-s + 0.114·20-s − 0.0343·21-s − 0.198·22-s − 0.251·23-s − 0.0576·24-s − 0.947·25-s − 0.647·26-s − 0.321·27-s − 0.105·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 2 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(36-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2 ^{s/2} \, \Gamma_{\C}(s+35/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(2\) |
| Conductor: | \(2\) |
| Sign: | $1$ |
| Analytic conductor: | \(15.5190\) |
| Root analytic conductor: | \(3.93941\) |
| Motivic weight: | \(35\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | yes |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((2,\ 2,\ (\ :35/2),\ 1)\) |
Particular Values
| \(L(18)\) | \(\approx\) | \(1.442420888\) |
| \(L(\frac12)\) | \(\approx\) | \(1.442420888\) |
| \(L(\frac{37}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 + p^{17} T \) |
| good | 3 | \( 1 - 4054972 p^{2} T + p^{35} T^{2} \) |
| 5 | \( 1 - 622513374 p^{4} T + p^{35} T^{2} \) | |
| 7 | \( 1 + 2646721826344 p^{2} T + p^{35} T^{2} \) | |
| 11 | \( 1 - 42694415520197532 p T + p^{35} T^{2} \) | |
| 13 | \( 1 - 168884893755185702 p^{2} T + p^{35} T^{2} \) | |
| 17 | \( 1 - \)\(31\!\cdots\!42\)\( p T + p^{35} T^{2} \) | |
| 19 | \( 1 - \)\(34\!\cdots\!80\)\( T + p^{35} T^{2} \) | |
| 23 | \( 1 + \)\(17\!\cdots\!72\)\( T + p^{35} T^{2} \) | |
| 29 | \( 1 - \)\(15\!\cdots\!30\)\( p T + p^{35} T^{2} \) | |
| 31 | \( 1 - \)\(62\!\cdots\!92\)\( p T + p^{35} T^{2} \) | |
| 37 | \( 1 - \)\(14\!\cdots\!54\)\( T + p^{35} T^{2} \) | |
| 41 | \( 1 - \)\(15\!\cdots\!02\)\( T + p^{35} T^{2} \) | |
| 43 | \( 1 - \)\(37\!\cdots\!08\)\( T + p^{35} T^{2} \) | |
| 47 | \( 1 + \)\(31\!\cdots\!76\)\( T + p^{35} T^{2} \) | |
| 53 | \( 1 - \)\(65\!\cdots\!98\)\( T + p^{35} T^{2} \) | |
| 59 | \( 1 + \)\(12\!\cdots\!60\)\( T + p^{35} T^{2} \) | |
| 61 | \( 1 - \)\(16\!\cdots\!02\)\( T + p^{35} T^{2} \) | |
| 67 | \( 1 - \)\(11\!\cdots\!64\)\( T + p^{35} T^{2} \) | |
| 71 | \( 1 + \)\(39\!\cdots\!48\)\( T + p^{35} T^{2} \) | |
| 73 | \( 1 + \)\(33\!\cdots\!22\)\( T + p^{35} T^{2} \) | |
| 79 | \( 1 - \)\(13\!\cdots\!20\)\( T + p^{35} T^{2} \) | |
| 83 | \( 1 + \)\(82\!\cdots\!32\)\( T + p^{35} T^{2} \) | |
| 89 | \( 1 + \)\(13\!\cdots\!90\)\( T + p^{35} T^{2} \) | |
| 97 | \( 1 - \)\(12\!\cdots\!74\)\( T + p^{35} T^{2} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−19.43264405561843700104301053472, −17.76035914036157685652583947239, −16.15923175722422351225122210702, −14.08180551289234999134128382457, −11.70117022291941455492072374679, −9.758481819581346611695344614498, −8.082781809545285380311551617528, −5.94861633959122458605162986850, −3.08498461902152729531977855320, −1.02367589470683279942751643648, 1.02367589470683279942751643648, 3.08498461902152729531977855320, 5.94861633959122458605162986850, 8.082781809545285380311551617528, 9.758481819581346611695344614498, 11.70117022291941455492072374679, 14.08180551289234999134128382457, 16.15923175722422351225122210702, 17.76035914036157685652583947239, 19.43264405561843700104301053472