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Properties

Label 2-5e4-25.16-c1-0-22
Degree $2$
Conductor $625$
Sign $0.992 + 0.125i$
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  + (1.30 + 0.951i)2-s + (−0.118 − 0.363i)3-s + (0.190 + 0.587i)4-s + (0.190 − 0.587i)6-s − 3·7-s + (0.690 − 2.12i)8-s + (2.30 − 1.67i)9-s + (2.42 + 1.76i)11-s + (0.190 − 0.138i)12-s + (3.92 − 2.85i)13-s + (−3.92 − 2.85i)14-s + (3.92 − 2.85i)16-s + (1.30 − 4.02i)17-s + 4.61·18-s + (−1.11 + 3.44i)19-s + ⋯
L(s)  = 1  + (0.925 + 0.672i)2-s + (−0.0681 − 0.209i)3-s + (0.0954 + 0.293i)4-s + (0.0779 − 0.239i)6-s − 1.13·7-s + (0.244 − 0.751i)8-s + (0.769 − 0.559i)9-s + (0.731 + 0.531i)11-s + (0.0551 − 0.0400i)12-s + (1.08 − 0.791i)13-s + (−1.04 − 0.762i)14-s + (0.981 − 0.713i)16-s + (0.317 − 0.977i)17-s + 1.08·18-s + (−0.256 + 0.789i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 + 0.125i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.992 + 0.125i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(625\)    =    \(5^{4}\)
Sign: $0.992 + 0.125i$
Analytic conductor: \(4.99065\)
Root analytic conductor: \(2.23397\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{625} (376, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 625,\ (\ :1/2),\ 0.992 + 0.125i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.30724 - 0.145159i\)
\(L(\frac12)\) \(\approx\) \(2.30724 - 0.145159i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.30 - 0.951i)T + (0.618 + 1.90i)T^{2} \)
3 \( 1 + (0.118 + 0.363i)T + (-2.42 + 1.76i)T^{2} \)
7 \( 1 + 3T + 7T^{2} \)
11 \( 1 + (-2.42 - 1.76i)T + (3.39 + 10.4i)T^{2} \)
13 \( 1 + (-3.92 + 2.85i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-1.30 + 4.02i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (1.11 - 3.44i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + (-1 - 0.726i)T + (7.10 + 21.8i)T^{2} \)
29 \( 1 + (-2.07 - 6.37i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-1.57 + 4.84i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (3 - 2.17i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (-2.42 + 1.76i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 9T + 43T^{2} \)
47 \( 1 + (2.57 + 7.91i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-1.42 - 4.39i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (3.35 - 2.43i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (-4.92 - 3.57i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + (4.28 - 13.1i)T + (-54.2 - 39.3i)T^{2} \)
71 \( 1 + (0.927 + 2.85i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (1.5 + 1.08i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (-0.163 - 0.502i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-0.145 + 0.449i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + (-10.8 - 7.88i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (-2.42 - 7.46i)T + (-78.4 + 57.0i)T^{2} \)
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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

−10.32299249772221501444530668869, −9.843264021293159291669217580404, −8.896962947202092700805068410303, −7.50684870469363063174175620980, −6.71130919378217510094103776679, −6.21748982557253763632163374867, −5.20585015792649268876063448044, −3.97619891696898043366659814084, −3.32049576332143229011766960735, −1.12748878967438659210837673130, 1.69067021465241943755698994487, 3.17431869568649634579266873876, 3.91238675239480184149859892589, 4.73386085956054442173962955082, 6.06711439472992073531994287364, 6.70686955884954272532537570585, 8.119710608375564835566954552144, 8.980893760999416854199523844911, 9.987124743526363282067235626110, 10.83999852156330219392582342845

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