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Properties

Label 2-5e4-25.14-c1-0-6
Degree $2$
Conductor $625$
Sign $0.425 + 0.904i$
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  + (−1.22 − 1.68i)2-s + (−2.09 − 0.679i)3-s + (−0.725 + 2.23i)4-s + (1.41 + 4.36i)6-s − 0.992i·7-s + (0.690 − 0.224i)8-s + (1.48 + 1.07i)9-s + (−1.61 + 1.17i)11-s + (3.03 − 4.17i)12-s + (−1.98 + 2.72i)13-s + (−1.67 + 1.21i)14-s + (2.57 + 1.87i)16-s + (2.75 − 0.894i)17-s − 3.82i·18-s + (0.798 + 2.45i)19-s + ⋯
L(s)  = 1  + (−0.866 − 1.19i)2-s + (−1.20 − 0.392i)3-s + (−0.362 + 1.11i)4-s + (0.578 + 1.78i)6-s − 0.375i·7-s + (0.244 − 0.0793i)8-s + (0.494 + 0.359i)9-s + (−0.487 + 0.354i)11-s + (0.876 − 1.20i)12-s + (−0.550 + 0.757i)13-s + (−0.447 + 0.325i)14-s + (0.643 + 0.467i)16-s + (0.667 − 0.216i)17-s − 0.901i·18-s + (0.183 + 0.563i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.425 + 0.904i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.425 + 0.904i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(625\)    =    \(5^{4}\)
Sign: $0.425 + 0.904i$
Analytic conductor: \(4.99065\)
Root analytic conductor: \(2.23397\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{625} (374, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 625,\ (\ :1/2),\ 0.425 + 0.904i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.375003 - 0.237984i\)
\(L(\frac12)\) \(\approx\) \(0.375003 - 0.237984i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.22 + 1.68i)T + (-0.618 + 1.90i)T^{2} \)
3 \( 1 + (2.09 + 0.679i)T + (2.42 + 1.76i)T^{2} \)
7 \( 1 + 0.992iT - 7T^{2} \)
11 \( 1 + (1.61 - 1.17i)T + (3.39 - 10.4i)T^{2} \)
13 \( 1 + (1.98 - 2.72i)T + (-4.01 - 12.3i)T^{2} \)
17 \( 1 + (-2.75 + 0.894i)T + (13.7 - 9.99i)T^{2} \)
19 \( 1 + (-0.798 - 2.45i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (-2.67 - 3.68i)T + (-7.10 + 21.8i)T^{2} \)
29 \( 1 + (-1.66 + 5.12i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-0.0421 - 0.129i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-1.26 + 1.73i)T + (-11.4 - 35.1i)T^{2} \)
41 \( 1 + (6.98 + 5.07i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 - 4.64iT - 43T^{2} \)
47 \( 1 + (9.44 + 3.06i)T + (38.0 + 27.6i)T^{2} \)
53 \( 1 + (-7.19 - 2.33i)T + (42.8 + 31.1i)T^{2} \)
59 \( 1 + (-3.97 - 2.89i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-2.24 + 1.62i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (-2.07 + 0.675i)T + (54.2 - 39.3i)T^{2} \)
71 \( 1 + (-2.97 + 9.17i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (0.456 + 0.627i)T + (-22.5 + 69.4i)T^{2} \)
79 \( 1 + (4.89 - 15.0i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-1.68 + 0.547i)T + (67.1 - 48.7i)T^{2} \)
89 \( 1 + (-11.7 + 8.52i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (-16.1 - 5.26i)T + (78.4 + 57.0i)T^{2} \)
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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

−10.40555566955395208391517822763, −9.964683346968012825490144564222, −9.027566954687306607720612716520, −7.85866535370182641797401197881, −7.01380720808490597725850321110, −5.89934571640463029771246352479, −4.92790710026169120289135667723, −3.50407881407620518036839902784, −2.08697595013944376760068630204, −0.835232834462728783299950408826, 0.56598992628096162920116903435, 3.03614822265795697209152811876, 4.95670752359846846028010577918, 5.43396532419785136809919818624, 6.30996683335781526355380150729, 7.13406948165509570111648142415, 8.142247404129537557301986626619, 8.824985458984264269568485483248, 9.996293942626557413011493611835, 10.42122383148563814973300878599

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