Dirichlet series
| L(s) = 1 | + 1.63e4·2-s + 4.78e6·3-s + 2.68e8·4-s + 6.06e9·5-s + 7.83e10·6-s + 9.04e11·7-s + 4.39e12·8-s − 4.57e13·9-s + 9.93e13·10-s + 2.34e15·11-s + 1.28e15·12-s + 1.60e16·13-s + 1.48e16·14-s + 2.90e16·15-s + 7.20e16·16-s − 5.35e17·17-s − 7.49e17·18-s − 4.50e18·19-s + 1.62e18·20-s + 4.32e18·21-s + 3.84e19·22-s + 6.05e18·23-s + 2.10e19·24-s − 1.49e20·25-s + 2.62e20·26-s − 5.47e20·27-s + 2.42e20·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.444·5-s + 0.408·6-s + 0.503·7-s + 0.353·8-s − 0.666·9-s + 0.314·10-s + 1.86·11-s + 0.288·12-s + 1.12·13-s + 0.356·14-s + 0.256·15-s + 1/4·16-s − 0.771·17-s − 0.471·18-s − 1.29·19-s + 0.222·20-s + 0.290·21-s + 1.31·22-s + 0.108·23-s + 0.204·24-s − 0.802·25-s + 0.797·26-s − 0.962·27-s + 0.251·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 2 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(30-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2 ^{s/2} \, \Gamma_{\C}(s+29/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(2\) |
| Conductor: | \(2\) |
| Sign: | $1$ |
| Analytic conductor: | \(10.6556\) |
| Root analytic conductor: | \(3.26429\) |
| Motivic weight: | \(29\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | yes |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((2,\ 2,\ (\ :29/2),\ 1)\) |
Particular Values
| \(L(15)\) | \(\approx\) | \(3.520517700\) |
| \(L(\frac12)\) | \(\approx\) | \(3.520517700\) |
| \(L(\frac{31}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 - p^{14} T \) |
| good | 3 | \( 1 - 177148 p^{3} T + p^{29} T^{2} \) |
| 5 | \( 1 - 48526734 p^{3} T + p^{29} T^{2} \) | |
| 7 | \( 1 - 18449364968 p^{2} T + p^{29} T^{2} \) | |
| 11 | \( 1 - 2348011244715132 T + p^{29} T^{2} \) | |
| 13 | \( 1 - 94693622446094 p^{2} T + p^{29} T^{2} \) | |
| 17 | \( 1 + 535853837930780718 T + p^{29} T^{2} \) | |
| 19 | \( 1 + 236865757052236660 p T + p^{29} T^{2} \) | |
| 23 | \( 1 - 263223728834583432 p T + p^{29} T^{2} \) | |
| 29 | \( 1 + \)\(23\!\cdots\!90\)\( T + p^{29} T^{2} \) | |
| 31 | \( 1 - \)\(45\!\cdots\!52\)\( T + p^{29} T^{2} \) | |
| 37 | \( 1 + \)\(50\!\cdots\!58\)\( T + p^{29} T^{2} \) | |
| 41 | \( 1 - \)\(15\!\cdots\!82\)\( T + p^{29} T^{2} \) | |
| 43 | \( 1 + \)\(21\!\cdots\!84\)\( T + p^{29} T^{2} \) | |
| 47 | \( 1 + \)\(88\!\cdots\!48\)\( T + p^{29} T^{2} \) | |
| 53 | \( 1 - \)\(19\!\cdots\!66\)\( T + p^{29} T^{2} \) | |
| 59 | \( 1 - \)\(41\!\cdots\!20\)\( T + p^{29} T^{2} \) | |
| 61 | \( 1 + \)\(63\!\cdots\!98\)\( T + p^{29} T^{2} \) | |
| 67 | \( 1 - \)\(15\!\cdots\!92\)\( T + p^{29} T^{2} \) | |
| 71 | \( 1 + \)\(45\!\cdots\!08\)\( T + p^{29} T^{2} \) | |
| 73 | \( 1 - \)\(38\!\cdots\!06\)\( T + p^{29} T^{2} \) | |
| 79 | \( 1 - \)\(40\!\cdots\!20\)\( T + p^{29} T^{2} \) | |
| 83 | \( 1 + \)\(12\!\cdots\!64\)\( T + p^{29} T^{2} \) | |
| 89 | \( 1 - \)\(10\!\cdots\!90\)\( T + p^{29} T^{2} \) | |
| 97 | \( 1 - \)\(10\!\cdots\!62\)\( T + p^{29} T^{2} \) | |
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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
−21.10777550561407898508515657169, −19.64784989359742791159791775155, −17.18615619207443291353083154968, −14.85638984224447553861345465074, −13.63619339434812474579912218204, −11.41154851897562389669197379941, −8.779970902532805434249063000958, −6.23144143952493656676120753846, −3.89473396346959387553655381787, −1.83375370370941610469032024100, 1.83375370370941610469032024100, 3.89473396346959387553655381787, 6.23144143952493656676120753846, 8.779970902532805434249063000958, 11.41154851897562389669197379941, 13.63619339434812474579912218204, 14.85638984224447553861345465074, 17.18615619207443291353083154968, 19.64784989359742791159791775155, 21.10777550561407898508515657169