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Properties

Label 2-5e4-25.11-c1-0-27
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  + (1.30 − 0.951i)2-s + (0.309 − 0.951i)3-s + (0.190 − 0.587i)4-s + (−0.499 − 1.53i)6-s + 0.618·7-s + (0.690 + 2.12i)8-s + (1.61 + 1.17i)9-s + (4.23 − 3.07i)11-s + (−0.5 − 0.363i)12-s + (−1.5 − 1.08i)13-s + (0.809 − 0.587i)14-s + (3.92 + 2.85i)16-s + (−1.61 − 4.97i)17-s + 3.23·18-s + (0.263 + 0.812i)19-s + ⋯
L(s)  = 1  + (0.925 − 0.672i)2-s + (0.178 − 0.549i)3-s + (0.0954 − 0.293i)4-s + (−0.204 − 0.628i)6-s + 0.233·7-s + (0.244 + 0.751i)8-s + (0.539 + 0.391i)9-s + (1.27 − 0.927i)11-s + (−0.144 − 0.104i)12-s + (−0.416 − 0.302i)13-s + (0.216 − 0.157i)14-s + (0.981 + 0.713i)16-s + (−0.392 − 1.20i)17-s + 0.762·18-s + (0.0605 + 0.186i)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: $\chi_{625} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 625,\ (\ :1/2),\ 0.425 + 0.904i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.31543 - 1.46941i\)
\(L(\frac12)\) \(\approx\) \(2.31543 - 1.46941i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.30 + 0.951i)T + (0.618 - 1.90i)T^{2} \)
3 \( 1 + (-0.309 + 0.951i)T + (-2.42 - 1.76i)T^{2} \)
7 \( 1 - 0.618T + 7T^{2} \)
11 \( 1 + (-4.23 + 3.07i)T + (3.39 - 10.4i)T^{2} \)
13 \( 1 + (1.5 + 1.08i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (1.61 + 4.97i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (-0.263 - 0.812i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (3.04 - 2.21i)T + (7.10 - 21.8i)T^{2} \)
29 \( 1 + (1.11 - 3.44i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (0.927 + 2.85i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-0.190 - 0.138i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (-0.618 - 0.449i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + 4.85T + 43T^{2} \)
47 \( 1 + (-0.190 + 0.587i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (1.07 - 3.30i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (-8.78 - 6.37i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (7.04 - 5.11i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (-1.47 - 4.53i)T + (-54.2 + 39.3i)T^{2} \)
71 \( 1 + (2.04 - 6.29i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-7.28 + 5.29i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (2.5 - 7.69i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (1.92 + 5.93i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + (-7.23 + 5.25i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (1.19 - 3.66i)T + (-78.4 - 57.0i)T^{2} \)
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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

−10.79622596491509055418777372534, −9.676836996863110767688379036829, −8.639979895132744480975057192718, −7.76613118476601152774415699121, −6.86994002775485692627927221299, −5.68777896347339740320310705244, −4.68608197160954099653057428744, −3.74779771447112335353816416776, −2.65600046576350178888865480925, −1.45311375910403269080389745421, 1.68932606569148120320967838498, 3.70947461787589027338849526120, 4.26796652563562304243492302631, 5.04596465763019445438571403720, 6.43363642145707491947028704427, 6.75686545919190870521053432729, 7.962833867909421057237548393341, 9.220692807777500869967382111234, 9.785238196425120033302251526423, 10.61608491730828214414593864276

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