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Properties

Label 2-5e4-25.9-c1-0-13
Degree $2$
Conductor $625$
Sign $0.425 - 0.904i$
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  + (0.107 − 0.148i)2-s + (1.39 − 0.454i)3-s + (0.607 + 1.87i)4-s + (0.0833 − 0.256i)6-s + 3.26i·7-s + (0.690 + 0.224i)8-s + (−0.674 + 0.489i)9-s + (−1.61 − 1.17i)11-s + (1.70 + 2.34i)12-s + (0.174 + 0.239i)13-s + (0.483 + 0.351i)14-s + (−3.07 + 2.23i)16-s + (4.91 + 1.59i)17-s + 0.152i·18-s + (−0.534 + 1.64i)19-s + ⋯
L(s)  = 1  + (0.0761 − 0.104i)2-s + (0.808 − 0.262i)3-s + (0.303 + 0.935i)4-s + (0.0340 − 0.104i)6-s + 1.23i·7-s + (0.244 + 0.0793i)8-s + (−0.224 + 0.163i)9-s + (−0.487 − 0.354i)11-s + (0.491 + 0.675i)12-s + (0.0483 + 0.0665i)13-s + (0.129 + 0.0938i)14-s + (−0.768 + 0.558i)16-s + (1.19 + 0.386i)17-s + 0.0359i·18-s + (−0.122 + 0.377i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.425 - 0.904i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.425 - 0.904i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(625\)    =    \(5^{4}\)
Sign: $0.425 - 0.904i$
Analytic conductor: \(4.99065\)
Root analytic conductor: \(2.23397\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{625} (249, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 625,\ (\ :1/2),\ 0.425 - 0.904i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.68823 + 1.07138i\)
\(L(\frac12)\) \(\approx\) \(1.68823 + 1.07138i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.107 + 0.148i)T + (-0.618 - 1.90i)T^{2} \)
3 \( 1 + (-1.39 + 0.454i)T + (2.42 - 1.76i)T^{2} \)
7 \( 1 - 3.26iT - 7T^{2} \)
11 \( 1 + (1.61 + 1.17i)T + (3.39 + 10.4i)T^{2} \)
13 \( 1 + (-0.174 - 0.239i)T + (-4.01 + 12.3i)T^{2} \)
17 \( 1 + (-4.91 - 1.59i)T + (13.7 + 9.99i)T^{2} \)
19 \( 1 + (0.534 - 1.64i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + (-0.516 + 0.711i)T + (-7.10 - 21.8i)T^{2} \)
29 \( 1 + (1.82 + 5.62i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-1.88 + 5.80i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-4.75 - 6.54i)T + (-11.4 + 35.1i)T^{2} \)
41 \( 1 + (-0.821 + 0.596i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 3.24iT - 43T^{2} \)
47 \( 1 + (-4.01 + 1.30i)T + (38.0 - 27.6i)T^{2} \)
53 \( 1 + (-7.70 + 2.50i)T + (42.8 - 31.1i)T^{2} \)
59 \( 1 + (-4.80 + 3.48i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (0.740 + 0.538i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + (6.55 + 2.12i)T + (54.2 + 39.3i)T^{2} \)
71 \( 1 + (1.84 + 5.67i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-5.19 + 7.14i)T + (-22.5 - 69.4i)T^{2} \)
79 \( 1 + (-2.39 - 7.38i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-13.8 - 4.48i)T + (67.1 + 48.7i)T^{2} \)
89 \( 1 + (-6.08 - 4.42i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (6.39 - 2.07i)T + (78.4 - 57.0i)T^{2} \)
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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

−10.92014093642117017868341611292, −9.713420455715311790434951059948, −8.756463813953429874801164297775, −8.101359688835410063824223259207, −7.65982606490884409063817667610, −6.24644644371205914754396279104, −5.35970238974976517637603023800, −3.83465381180779367684435392312, −2.83743101365522621841638051095, −2.16079881493560758230595424895, 1.01558198900896374371762983638, 2.62146289288296814702160782195, 3.75430245919133842705565509700, 4.88246670727134565260815334959, 5.86721655554920213900979372972, 7.07561679854809404218738334262, 7.63658076862006260315629643141, 8.848299450545260192226962759996, 9.689799427981636068923929321799, 10.36409151496140556157880697354

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