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Properties

Label 2-5e4-25.6-c1-0-11
Degree $2$
Conductor $625$
Sign $0.268 + 0.963i$
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  + (−0.823 − 2.53i)2-s + (0.614 − 0.446i)3-s + (−4.12 + 2.99i)4-s + (−1.63 − 1.18i)6-s + 2.04·7-s + (6.68 + 4.85i)8-s + (−0.748 + 2.30i)9-s + (0.416 + 1.28i)11-s + (−1.19 + 3.68i)12-s + (−0.407 + 1.25i)13-s + (−1.68 − 5.17i)14-s + (3.65 − 11.2i)16-s + (3.30 + 2.40i)17-s + 6.45·18-s + (3.95 + 2.87i)19-s + ⋯
L(s)  = 1  + (−0.582 − 1.79i)2-s + (0.354 − 0.257i)3-s + (−2.06 + 1.49i)4-s + (−0.668 − 0.485i)6-s + 0.771·7-s + (2.36 + 1.71i)8-s + (−0.249 + 0.768i)9-s + (0.125 + 0.386i)11-s + (−0.345 + 1.06i)12-s + (−0.113 + 0.348i)13-s + (−0.449 − 1.38i)14-s + (0.913 − 2.81i)16-s + (0.801 + 0.582i)17-s + 1.52·18-s + (0.906 + 0.658i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.268 + 0.963i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.268 + 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(625\)    =    \(5^{4}\)
Sign: $0.268 + 0.963i$
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: \((2,\ 625,\ (\ :1/2),\ 0.268 + 0.963i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.914829 - 0.694393i\)
\(L(\frac12)\) \(\approx\) \(0.914829 - 0.694393i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.823 + 2.53i)T + (-1.61 + 1.17i)T^{2} \)
3 \( 1 + (-0.614 + 0.446i)T + (0.927 - 2.85i)T^{2} \)
7 \( 1 - 2.04T + 7T^{2} \)
11 \( 1 + (-0.416 - 1.28i)T + (-8.89 + 6.46i)T^{2} \)
13 \( 1 + (0.407 - 1.25i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (-3.30 - 2.40i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-3.95 - 2.87i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (-0.845 - 2.60i)T + (-18.6 + 13.5i)T^{2} \)
29 \( 1 + (-3.73 + 2.71i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (5.79 + 4.20i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-2.67 + 8.22i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (3.12 - 9.60i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 2.43T + 43T^{2} \)
47 \( 1 + (-6.12 + 4.45i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (0.502 - 0.365i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (3.50 - 10.7i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-0.200 - 0.615i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + (-8.84 - 6.42i)T + (20.7 + 63.7i)T^{2} \)
71 \( 1 + (-2.38 + 1.73i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (4.18 + 12.8i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (1.52 - 1.11i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-1.92 - 1.39i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + (-2.26 - 6.97i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (-4.69 + 3.40i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.47117698069378217291169008075, −9.742347184586226475615121757873, −8.911043092910749380013645954264, −7.979322516687704479999858031518, −7.55825792227012660081637546350, −5.49171279790869144434309633445, −4.43574049834799513058448684375, −3.38973945117652157385733723659, −2.21675778125151810214622795758, −1.38575423585305901543249236859, 0.879908455928177496957124038651, 3.34098791039205686685219427302, 4.78910510303717732800456723064, 5.41864450334705844162665648712, 6.49497307258749898342191563055, 7.30248910559700784200082758762, 8.163308524014723019228708831754, 8.813701814635939723422909001916, 9.497506203682107766318803649519, 10.34053494205061431324283906610

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