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Properties

Label 2-5e4-25.6-c1-0-13
Degree $2$
Conductor $625$
Sign $0.146 - 0.989i$
Analytic cond. $4.99065$
Root an. cond. $2.23397$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.154 + 0.476i)2-s + (−2.50 + 1.81i)3-s + (1.41 − 1.02i)4-s + (−1.25 − 0.910i)6-s + 0.0237·7-s + (1.51 + 1.10i)8-s + (2.02 − 6.24i)9-s + (1.10 + 3.40i)11-s + (−1.67 + 5.14i)12-s + (1.16 − 3.59i)13-s + (0.00368 + 0.0113i)14-s + (0.790 − 2.43i)16-s + (2.93 + 2.12i)17-s + 3.29·18-s + (1.96 + 1.42i)19-s + ⋯
L(s)  = 1  + (0.109 + 0.336i)2-s + (−1.44 + 1.04i)3-s + (0.707 − 0.514i)4-s + (−0.511 − 0.371i)6-s + 0.00899·7-s + (0.537 + 0.390i)8-s + (0.676 − 2.08i)9-s + (0.333 + 1.02i)11-s + (−0.482 + 1.48i)12-s + (0.323 − 0.995i)13-s + (0.000984 + 0.00302i)14-s + (0.197 − 0.607i)16-s + (0.710 + 0.516i)17-s + 0.775·18-s + (0.451 + 0.327i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.146 - 0.989i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.146 - 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(625\)    =    \(5^{4}\)
Sign: $0.146 - 0.989i$
Analytic conductor: \(4.99065\)
Root analytic conductor: \(2.23397\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{625} (126, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 625,\ (\ :1/2),\ 0.146 - 0.989i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.932494 + 0.804907i\)
\(L(\frac12)\) \(\approx\) \(0.932494 + 0.804907i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
good2 \( 1 + (-0.154 - 0.476i)T + (-1.61 + 1.17i)T^{2} \)
3 \( 1 + (2.50 - 1.81i)T + (0.927 - 2.85i)T^{2} \)
7 \( 1 - 0.0237T + 7T^{2} \)
11 \( 1 + (-1.10 - 3.40i)T + (-8.89 + 6.46i)T^{2} \)
13 \( 1 + (-1.16 + 3.59i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (-2.93 - 2.12i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-1.96 - 1.42i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (-0.530 - 1.63i)T + (-18.6 + 13.5i)T^{2} \)
29 \( 1 + (3.12 - 2.26i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-4.86 - 3.53i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (0.114 - 0.351i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (2.41 - 7.42i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 0.174T + 43T^{2} \)
47 \( 1 + (6.31 - 4.59i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (-7.25 + 5.27i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (1.37 - 4.23i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-2.84 - 8.76i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + (-3.61 - 2.62i)T + (20.7 + 63.7i)T^{2} \)
71 \( 1 + (-7.84 + 5.69i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (1.22 + 3.76i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-7.83 + 5.69i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-7.24 - 5.26i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + (5.25 + 16.1i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (2.23 - 1.62i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.67937186383175854999451891630, −10.14190803607685537506829450286, −9.562776680807199387913509520668, −7.986517045563230250356587222245, −6.90708084351588697380436339308, −6.10677623791957227726045257295, −5.37836137456050981547779391300, −4.68300794483714486367996645673, −3.38239430558528254099072496237, −1.30064129062877623493258137544, 0.927052270515350314768999706134, 2.17911362410869480069763892830, 3.66709910715840177034708949455, 5.06875727912272547017711909496, 6.12580010252085538289059483035, 6.70370888344442810998449873987, 7.48414142437442562452373593965, 8.396999086443727892177559296631, 9.820154646582532306787683733826, 10.99633719781409094941092676647

Graph of the $Z$-function along the critical line