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Properties

Label 2-5e4-25.16-c1-0-19
Degree $2$
Conductor $625$
Sign $0.0209 - 0.999i$
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  + (2.15 + 1.56i)2-s + (−0.234 − 0.721i)3-s + (1.57 + 4.85i)4-s + (0.625 − 1.92i)6-s + 2.04·7-s + (−2.55 + 7.86i)8-s + (1.96 − 1.42i)9-s + (−1.09 − 0.792i)11-s + (3.13 − 2.27i)12-s + (1.06 − 0.775i)13-s + (4.40 + 3.19i)14-s + (−9.56 + 6.95i)16-s + (−1.26 + 3.88i)17-s + 6.45·18-s + (−1.51 + 4.64i)19-s + ⋯
L(s)  = 1  + (1.52 + 1.10i)2-s + (−0.135 − 0.416i)3-s + (0.788 + 2.42i)4-s + (0.255 − 0.785i)6-s + 0.771·7-s + (−0.903 + 2.77i)8-s + (0.653 − 0.474i)9-s + (−0.328 − 0.239i)11-s + (0.904 − 0.657i)12-s + (0.296 − 0.215i)13-s + (1.17 + 0.854i)14-s + (−2.39 + 1.73i)16-s + (−0.306 + 0.942i)17-s + 1.52·18-s + (−0.346 + 1.06i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0209 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0209 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(625\)    =    \(5^{4}\)
Sign: $0.0209 - 0.999i$
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: \((2,\ 625,\ (\ :1/2),\ 0.0209 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.48558 + 2.43406i\)
\(L(\frac12)\) \(\approx\) \(2.48558 + 2.43406i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.15 - 1.56i)T + (0.618 + 1.90i)T^{2} \)
3 \( 1 + (0.234 + 0.721i)T + (-2.42 + 1.76i)T^{2} \)
7 \( 1 - 2.04T + 7T^{2} \)
11 \( 1 + (1.09 + 0.792i)T + (3.39 + 10.4i)T^{2} \)
13 \( 1 + (-1.06 + 0.775i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (1.26 - 3.88i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (1.51 - 4.64i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + (2.21 + 1.60i)T + (7.10 + 21.8i)T^{2} \)
29 \( 1 + (1.42 + 4.39i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-2.21 + 6.80i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (6.99 - 5.08i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (-8.17 + 5.93i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 2.43T + 43T^{2} \)
47 \( 1 + (2.33 + 7.20i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-0.191 - 0.590i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (-9.17 + 6.66i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (0.523 + 0.380i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + (3.38 - 10.4i)T + (-54.2 - 39.3i)T^{2} \)
71 \( 1 + (0.912 + 2.80i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-10.9 - 7.95i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (-0.583 - 1.79i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (0.733 - 2.25i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + (5.93 + 4.31i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (1.79 + 5.51i)T + (-78.4 + 57.0i)T^{2} \)
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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

−11.24376260980153875773037567834, −10.06802786007193265956909065063, −8.375717060431021513261974818106, −8.009413606734150913463182285140, −7.00565949109662279101179495412, −6.17442010396192962000876472405, −5.53152196277037094229571415684, −4.32608366278836485565716455678, −3.71089900643785780105089010996, −2.06956332681348278142918572623, 1.52574057981633482419162874480, 2.62845431821111200384828755758, 3.87085978297438778146723780881, 4.88236608580364639540783937581, 5.08065595653221829616475945579, 6.47712227055950762662375000714, 7.48442787239382438666409587357, 9.032748174587496906220115536209, 9.962266856869548710988327874596, 10.86593773328441789768360296556

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