Error: table mf_hecke_newspace_traces does not exist
Dirichlet series
| L(s) = 1 | + (−1.24 − 1.24i)2-s + 1.11i·4-s + 5-s − 2.67·7-s + (−1.10 + 1.10i)8-s + (−1.24 − 1.24i)10-s + (0.991 + 0.991i)11-s − 2.00i·13-s + (3.34 + 3.34i)14-s + 4.98·16-s + (2.41 + 2.41i)17-s + (4.17 − 4.17i)19-s + 1.11i·20-s − 2.47i·22-s + 0.783i·23-s + ⋯ |
| L(s) = 1 | + (−0.882 − 0.882i)2-s + 0.559i·4-s + 0.447·5-s − 1.01·7-s + (−0.389 + 0.389i)8-s + (−0.394 − 0.394i)10-s + (0.298 + 0.298i)11-s − 0.554i·13-s + (0.893 + 0.893i)14-s + 1.24·16-s + (0.585 + 0.585i)17-s + (0.956 − 0.956i)19-s + 0.250i·20-s − 0.527i·22-s + 0.163i·23-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.784 + 0.620i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.784 + 0.620i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Invariants
| Degree: | \(2\) |
| Conductor: | \(1305\) = \(3^{2} \cdot 5 \cdot 29\) |
| Sign: | $-0.784 + 0.620i$ |
| Analytic conductor: | \(10.4204\) |
| Root analytic conductor: | \(3.22807\) |
| Motivic weight: | \(1\) |
| Rational: | no |
| Arithmetic: | yes |
| Character: | $\chi_{1305} (1061, \cdot )$ |
| Primitive: | yes |
| Self-dual: | no |
| Analytic rank: | \(0\) |
| Selberg data: | \((2,\ 1305,\ (\ :1/2),\ -0.784 + 0.620i)\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.7656229281\) |
| \(L(\frac12)\) | \(\approx\) | \(0.7656229281\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - T \) | |
| 29 | \( 1 + (5.22 - 1.30i)T \) | |
| good | 2 | \( 1 + (1.24 + 1.24i)T + 2iT^{2} \) |
| 7 | \( 1 + 2.67T + 7T^{2} \) | |
| 11 | \( 1 + (-0.991 - 0.991i)T + 11iT^{2} \) | |
| 13 | \( 1 + 2.00iT - 13T^{2} \) | |
| 17 | \( 1 + (-2.41 - 2.41i)T + 17iT^{2} \) | |
| 19 | \( 1 + (-4.17 + 4.17i)T - 19iT^{2} \) | |
| 23 | \( 1 - 0.783iT - 23T^{2} \) | |
| 31 | \( 1 + (-3.26 + 3.26i)T - 31iT^{2} \) | |
| 37 | \( 1 + (6.88 + 6.88i)T + 37iT^{2} \) | |
| 41 | \( 1 + (-3.90 + 3.90i)T - 41iT^{2} \) | |
| 43 | \( 1 + (-1.51 + 1.51i)T - 43iT^{2} \) | |
| 47 | \( 1 + (1.81 - 1.81i)T - 47iT^{2} \) | |
| 53 | \( 1 - 0.590iT - 53T^{2} \) | |
| 59 | \( 1 + 5.39iT - 59T^{2} \) | |
| 61 | \( 1 + (-4.38 + 4.38i)T - 61iT^{2} \) | |
| 67 | \( 1 + 12.5iT - 67T^{2} \) | |
| 71 | \( 1 - 9.09T + 71T^{2} \) | |
| 73 | \( 1 + (-0.687 - 0.687i)T + 73iT^{2} \) | |
| 79 | \( 1 + (6.86 - 6.86i)T - 79iT^{2} \) | |
| 83 | \( 1 + 4.93iT - 83T^{2} \) | |
| 89 | \( 1 + (8.02 + 8.02i)T + 89iT^{2} \) | |
| 97 | \( 1 + (1.19 + 1.19i)T + 97iT^{2} \) | |
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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
−9.525001734606603144684261726904, −8.926254934450976069895020794999, −7.898881087883597188730056115180, −6.97939466817907181292480635146, −5.95493007143683436931178043517, −5.26465895196060456721844718395, −3.67930821299809520322922039998, −2.88669468437635246514456384150, −1.79944656114709311826803323458, −0.48901016518655505769443980141, 1.17573496706507530946572226617, 2.93879716527359298291044463144, 3.77790899719736979629408229479, 5.31260985271184404807332551240, 6.11655008630435588811102910200, 6.76646712863117863400988665037, 7.48063566634152254321204239865, 8.381021338425958879430418861219, 9.152392706027714626020492610442, 9.795565510974734169510750900871