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Properties

Label 2-5e4-25.6-c1-0-31
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

\(L(1)\) \(\approx\) \(1.67013 - 1.93486i\)
\(L(\frac12)\) \(\approx\) \(1.67013 - 1.93486i\)
\(L(\frac{3}{2})\) 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

−10.10925032486315883596901045621, −9.426461999306728814353827425350, −8.751841802387131634163320381923, −7.53943154893331517655093803151, −6.99716202858313030303803167538, −6.30604629163159457590874414344, −4.57233943517517898775877589173, −3.18116734859371458088127441558, −2.21756852317077890486040940028, −1.46122195823279898194335520878, 2.29979183367674778215056605138, 3.20638978936293381752918270076, 3.92238522769031295985093767050, 5.29535740589529118624519895461, 6.48106374240408526393441097683, 7.75938362346034867993874810436, 8.175366326460054644864808264805, 8.982699813553159656622502133448, 9.778867829767162072554864161350, 10.77033881831090661610269421564

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