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Properties

Label 2-5e4-25.4-c1-0-8
Degree $2$
Conductor $625$
Sign $0.0627 - 0.998i$
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  + (−1.07 − 0.350i)2-s + (−1.52 + 2.10i)3-s + (−0.578 − 0.420i)4-s + (2.38 − 1.73i)6-s − 0.407i·7-s + (1.80 + 2.48i)8-s + (−1.16 − 3.58i)9-s + (0.618 − 1.90i)11-s + (1.76 − 0.574i)12-s + (0.666 − 0.216i)13-s + (−0.142 + 0.439i)14-s + (−0.636 − 1.95i)16-s + (0.930 + 1.28i)17-s + 4.27i·18-s + (4.00 − 2.90i)19-s + ⋯
L(s)  = 1  + (−0.762 − 0.247i)2-s + (−0.883 + 1.21i)3-s + (−0.289 − 0.210i)4-s + (0.974 − 0.708i)6-s − 0.153i·7-s + (0.639 + 0.880i)8-s + (−0.388 − 1.19i)9-s + (0.186 − 0.573i)11-s + (0.510 − 0.165i)12-s + (0.184 − 0.0600i)13-s + (−0.0381 + 0.117i)14-s + (−0.159 − 0.489i)16-s + (0.225 + 0.310i)17-s + 1.00i·18-s + (0.918 − 0.667i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0627 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0627 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(625\)    =    \(5^{4}\)
Sign: $0.0627 - 0.998i$
Analytic conductor: \(4.99065\)
Root analytic conductor: \(2.23397\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{625} (124, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 625,\ (\ :1/2),\ 0.0627 - 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.394384 + 0.370352i\)
\(L(\frac12)\) \(\approx\) \(0.394384 + 0.370352i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.07 + 0.350i)T + (1.61 + 1.17i)T^{2} \)
3 \( 1 + (1.52 - 2.10i)T + (-0.927 - 2.85i)T^{2} \)
7 \( 1 + 0.407iT - 7T^{2} \)
11 \( 1 + (-0.618 + 1.90i)T + (-8.89 - 6.46i)T^{2} \)
13 \( 1 + (-0.666 + 0.216i)T + (10.5 - 7.64i)T^{2} \)
17 \( 1 + (-0.930 - 1.28i)T + (-5.25 + 16.1i)T^{2} \)
19 \( 1 + (-4.00 + 2.90i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (-1.14 - 0.371i)T + (18.6 + 13.5i)T^{2} \)
29 \( 1 + (-4.45 - 3.23i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (6.63 - 4.82i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (4.88 - 1.58i)T + (29.9 - 21.7i)T^{2} \)
41 \( 1 + (-2.22 - 6.86i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 9.16iT - 43T^{2} \)
47 \( 1 + (-0.748 + 1.03i)T + (-14.5 - 44.6i)T^{2} \)
53 \( 1 + (2.98 - 4.10i)T + (-16.3 - 50.4i)T^{2} \)
59 \( 1 + (-2.00 - 6.18i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (2.91 - 8.95i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + (1.81 + 2.49i)T + (-20.7 + 63.7i)T^{2} \)
71 \( 1 + (5.55 + 4.03i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-0.518 - 0.168i)T + (59.0 + 42.9i)T^{2} \)
79 \( 1 + (-4.43 - 3.22i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (0.572 + 0.788i)T + (-25.6 + 78.9i)T^{2} \)
89 \( 1 + (-0.700 + 2.15i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (8.94 - 12.3i)T + (-29.9 - 92.2i)T^{2} \)
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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

−10.70715771112157607495304410341, −10.04981354655604994501457132952, −9.246301532835035180101752011333, −8.634596514368315122171819776505, −7.41204099734819724537438360927, −6.05154922310540930199086821777, −5.21231424961972374858754550235, −4.48828139473731481697998454802, −3.24395101422803210159536885261, −1.10421441499899767923854050169, 0.55167843555714431159086209040, 1.86382862361576112001884538024, 3.75185613413115827505057453672, 5.13737792206891357103110115373, 6.09220800293337351184452617775, 7.18763992615509847660230053689, 7.47738368277048711112048705403, 8.533211183878451282023069094476, 9.449191429496340574893863095651, 10.30725451549511279776228155820

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