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Properties

Label 2-26e2-676.659-c0-0-0
Degree $2$
Conductor $676$
Sign $0.902 - 0.431i$
Analytic cond. $0.337367$
Root an. cond. $0.580833$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.200 − 0.979i)2-s + (−0.919 + 0.391i)4-s + (−1.84 + 0.453i)5-s + (0.568 + 0.822i)8-s + (0.948 + 0.316i)9-s + (0.813 + 1.71i)10-s + (0.428 + 0.903i)13-s + (0.692 − 0.721i)16-s + (−0.788 + 1.66i)17-s + (0.120 − 0.992i)18-s + (1.51 − 1.13i)20-s + (2.30 − 1.20i)25-s + (0.799 − 0.600i)26-s + (0.253 + 1.23i)29-s + (−0.845 − 0.534i)32-s + ⋯
L(s)  = 1  + (−0.200 − 0.979i)2-s + (−0.919 + 0.391i)4-s + (−1.84 + 0.453i)5-s + (0.568 + 0.822i)8-s + (0.948 + 0.316i)9-s + (0.813 + 1.71i)10-s + (0.428 + 0.903i)13-s + (0.692 − 0.721i)16-s + (−0.788 + 1.66i)17-s + (0.120 − 0.992i)18-s + (1.51 − 1.13i)20-s + (2.30 − 1.20i)25-s + (0.799 − 0.600i)26-s + (0.253 + 1.23i)29-s + (−0.845 − 0.534i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 676 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.902 - 0.431i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 676 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.902 - 0.431i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(676\)    =    \(2^{2} \cdot 13^{2}\)
Sign: $0.902 - 0.431i$
Analytic conductor: \(0.337367\)
Root analytic conductor: \(0.580833\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{676} (659, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 676,\ (\ :0),\ 0.902 - 0.431i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5218361060\)
\(L(\frac12)\) \(\approx\) \(0.5218361060\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.200 + 0.979i)T \)
13 \( 1 + (-0.428 - 0.903i)T \)
good3 \( 1 + (-0.948 - 0.316i)T^{2} \)
5 \( 1 + (1.84 - 0.453i)T + (0.885 - 0.464i)T^{2} \)
7 \( 1 + (0.632 + 0.774i)T^{2} \)
11 \( 1 + (-0.799 + 0.600i)T^{2} \)
17 \( 1 + (0.788 - 1.66i)T + (-0.632 - 0.774i)T^{2} \)
19 \( 1 + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.5 + 0.866i)T^{2} \)
29 \( 1 + (-0.253 - 1.23i)T + (-0.919 + 0.391i)T^{2} \)
31 \( 1 + (-0.568 - 0.822i)T^{2} \)
37 \( 1 + (0.470 - 0.297i)T + (0.428 - 0.903i)T^{2} \)
41 \( 1 + (0.394 + 0.0641i)T + (0.948 + 0.316i)T^{2} \)
43 \( 1 + (-0.428 - 0.903i)T^{2} \)
47 \( 1 + (0.970 - 0.239i)T^{2} \)
53 \( 1 + (-0.487 - 0.705i)T + (-0.354 + 0.935i)T^{2} \)
59 \( 1 + (0.0402 - 0.999i)T^{2} \)
61 \( 1 + (1.96 + 0.158i)T + (0.987 + 0.160i)T^{2} \)
67 \( 1 + (-0.692 - 0.721i)T^{2} \)
71 \( 1 + (0.200 + 0.979i)T^{2} \)
73 \( 1 + (-0.748 - 0.663i)T + (0.120 + 0.992i)T^{2} \)
79 \( 1 + (0.970 - 0.239i)T^{2} \)
83 \( 1 + (0.748 + 0.663i)T^{2} \)
89 \( 1 + (-0.970 + 1.68i)T + (-0.5 - 0.866i)T^{2} \)
97 \( 1 + (-0.492 - 1.70i)T + (-0.845 + 0.534i)T^{2} \)
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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

−10.81525591722222219235854187010, −10.30595331693414840521113489010, −8.928154516152242809839321580723, −8.330828995143342027177133220877, −7.46522546090941968875112669320, −6.61089187864149214021876905786, −4.69963363758106114653406785360, −4.07410479092954112068428929806, −3.33261905374705925155469010198, −1.69391392487600350708610890527, 0.67943031994229321658770323688, 3.42687843627754676949628381782, 4.38351466985175787968352546227, 5.00875252314458896791239073439, 6.46101780496884349042030874794, 7.37685375099937595245136927428, 7.82305868251167876673663595684, 8.716622269081247041696251323241, 9.490775710820976121305239458975, 10.58365759183390262927293237254

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