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Properties

Label 2-5e4-25.16-c1-0-9
Degree $2$
Conductor $625$
Sign $-0.944 - 0.328i$
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  + (1.88 + 1.36i)2-s + (0.711 + 2.18i)3-s + (1.05 + 3.24i)4-s + (−1.65 + 5.09i)6-s − 3.59·7-s + (−1.01 + 3.12i)8-s + (−1.86 + 1.35i)9-s + (0.402 + 0.292i)11-s + (−6.35 + 4.61i)12-s + (2.14 − 1.55i)13-s + (−6.76 − 4.91i)14-s + (−0.656 + 0.477i)16-s + (−1.57 + 4.85i)17-s − 5.35·18-s + (−0.305 + 0.938i)19-s + ⋯
L(s)  = 1  + (1.33 + 0.966i)2-s + (0.410 + 1.26i)3-s + (0.526 + 1.62i)4-s + (−0.675 + 2.07i)6-s − 1.35·7-s + (−0.358 + 1.10i)8-s + (−0.620 + 0.450i)9-s + (0.121 + 0.0882i)11-s + (−1.83 + 1.33i)12-s + (0.594 − 0.431i)13-s + (−1.80 − 1.31i)14-s + (−0.164 + 0.119i)16-s + (−0.382 + 1.17i)17-s − 1.26·18-s + (−0.0699 + 0.215i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.944 - 0.328i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.944 - 0.328i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(625\)    =    \(5^{4}\)
Sign: $-0.944 - 0.328i$
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: \((2,\ 625,\ (\ :1/2),\ -0.944 - 0.328i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.507492 + 3.00047i\)
\(L(\frac12)\) \(\approx\) \(0.507492 + 3.00047i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.88 - 1.36i)T + (0.618 + 1.90i)T^{2} \)
3 \( 1 + (-0.711 - 2.18i)T + (-2.42 + 1.76i)T^{2} \)
7 \( 1 + 3.59T + 7T^{2} \)
11 \( 1 + (-0.402 - 0.292i)T + (3.39 + 10.4i)T^{2} \)
13 \( 1 + (-2.14 + 1.55i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (1.57 - 4.85i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (0.305 - 0.938i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + (-5.18 - 3.76i)T + (7.10 + 21.8i)T^{2} \)
29 \( 1 + (1.72 + 5.29i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-1.87 + 5.75i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (3.71 - 2.70i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (-2.32 + 1.68i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 - 9.48T + 43T^{2} \)
47 \( 1 + (1.65 + 5.10i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-0.0950 - 0.292i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (-1.02 + 0.744i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (-5.03 - 3.65i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + (-1.63 + 5.02i)T + (-54.2 - 39.3i)T^{2} \)
71 \( 1 + (0.0469 + 0.144i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (12.0 + 8.74i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (5.12 + 15.7i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (4.51 - 13.8i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + (9.20 + 6.69i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (0.262 + 0.807i)T + (-78.4 + 57.0i)T^{2} \)
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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

−10.90495468543700961285784304366, −10.02405659033061651854309040683, −9.290174821893255459244493214036, −8.310169190622858819230579931629, −7.15914575237554847373480225182, −6.20216436988530256878090265978, −5.57867765827121600383834603433, −4.32218262801484874128497613802, −3.74901275231829441443841538546, −2.98412188015498957520617083238, 1.14759162953092402370397420764, 2.56553509681758768600576010165, 3.13851670722769533016925563567, 4.36464337103662722358197181059, 5.59217668663920353846327760903, 6.68180428454241562527372636546, 7.03316768963280290666266456497, 8.581008016302275528240114658077, 9.425488446447503893172929767169, 10.58855350014832457008874681072

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