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Properties

Label 2-26e2-676.211-c0-0-0
Degree $2$
Conductor $676$
Sign $0.982 - 0.184i$
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  + (−0.0402 − 0.999i)2-s + (−0.996 + 0.0804i)4-s + (0.599 + 1.58i)5-s + (0.120 + 0.992i)8-s + (−0.845 + 0.534i)9-s + (1.55 − 0.662i)10-s + (−0.919 + 0.391i)13-s + (0.987 − 0.160i)16-s + (1.83 + 0.781i)17-s + (0.568 + 0.822i)18-s + (−0.724 − 1.52i)20-s + (−1.39 + 1.23i)25-s + (0.428 + 0.903i)26-s + (−0.0557 − 1.38i)29-s + (−0.200 − 0.979i)32-s + ⋯
L(s)  = 1  + (−0.0402 − 0.999i)2-s + (−0.996 + 0.0804i)4-s + (0.599 + 1.58i)5-s + (0.120 + 0.992i)8-s + (−0.845 + 0.534i)9-s + (1.55 − 0.662i)10-s + (−0.919 + 0.391i)13-s + (0.987 − 0.160i)16-s + (1.83 + 0.781i)17-s + (0.568 + 0.822i)18-s + (−0.724 − 1.52i)20-s + (−1.39 + 1.23i)25-s + (0.428 + 0.903i)26-s + (−0.0557 − 1.38i)29-s + (−0.200 − 0.979i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 676 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.982 - 0.184i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 676 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.982 - 0.184i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(676\)    =    \(2^{2} \cdot 13^{2}\)
Sign: $0.982 - 0.184i$
Analytic conductor: \(0.337367\)
Root analytic conductor: \(0.580833\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{676} (211, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 676,\ (\ :0),\ 0.982 - 0.184i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8410504150\)
\(L(\frac12)\) \(\approx\) \(0.8410504150\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.0402 + 0.999i)T \)
13 \( 1 + (0.919 - 0.391i)T \)
good3 \( 1 + (0.845 - 0.534i)T^{2} \)
5 \( 1 + (-0.599 - 1.58i)T + (-0.748 + 0.663i)T^{2} \)
7 \( 1 + (-0.692 - 0.721i)T^{2} \)
11 \( 1 + (-0.428 - 0.903i)T^{2} \)
17 \( 1 + (-1.83 - 0.781i)T + (0.692 + 0.721i)T^{2} \)
19 \( 1 + (0.5 + 0.866i)T^{2} \)
23 \( 1 + (0.5 - 0.866i)T^{2} \)
29 \( 1 + (0.0557 + 1.38i)T + (-0.996 + 0.0804i)T^{2} \)
31 \( 1 + (-0.120 - 0.992i)T^{2} \)
37 \( 1 + (-0.253 + 1.23i)T + (-0.919 - 0.391i)T^{2} \)
41 \( 1 + (0.0224 + 0.0773i)T + (-0.845 + 0.534i)T^{2} \)
43 \( 1 + (0.919 - 0.391i)T^{2} \)
47 \( 1 + (0.354 + 0.935i)T^{2} \)
53 \( 1 + (0.221 + 1.82i)T + (-0.970 + 0.239i)T^{2} \)
59 \( 1 + (-0.948 - 0.316i)T^{2} \)
61 \( 1 + (-0.444 - 0.334i)T + (0.278 + 0.960i)T^{2} \)
67 \( 1 + (-0.987 - 0.160i)T^{2} \)
71 \( 1 + (0.0402 + 0.999i)T^{2} \)
73 \( 1 + (0.885 - 0.464i)T + (0.568 - 0.822i)T^{2} \)
79 \( 1 + (0.354 + 0.935i)T^{2} \)
83 \( 1 + (-0.885 + 0.464i)T^{2} \)
89 \( 1 + (-0.354 - 0.614i)T + (-0.5 + 0.866i)T^{2} \)
97 \( 1 + (-0.946 - 1.15i)T + (-0.200 + 0.979i)T^{2} \)
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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

−10.65002508277248748599034874075, −10.08159066880676102089985176689, −9.435944129421183523177537086868, −8.136042210871369906163480965328, −7.40157706826179263978525449561, −6.06272582333164967952639350210, −5.34999099503181186154520166341, −3.84006138361731653205649285137, −2.84296013993943377131664973966, −2.09455500361683792344951529925, 1.03318018250164999106546499845, 3.20518469549545095306049647580, 4.71584558433736338664715881759, 5.34790526237039718606123495475, 5.95383845245559629117781028265, 7.26951894820061823402286542766, 8.143399495839892942374762922316, 8.901658652803614369095518583295, 9.529106987952403493660018103781, 10.21395302151614378511252602306

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