Properties

Label 2-26e2-52.23-c0-0-0
Degree $2$
Conductor $676$
Sign $-0.265 - 0.964i$
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  + (−0.866 + 0.5i)2-s + (0.499 − 0.866i)4-s + i·5-s + 0.999i·8-s + (−0.5 + 0.866i)9-s + (−0.5 − 0.866i)10-s + (−0.5 − 0.866i)16-s + (−0.5 + 0.866i)17-s − 0.999i·18-s + (0.866 + 0.499i)20-s + (0.5 + 0.866i)29-s + (0.866 + 0.499i)32-s − 0.999i·34-s + (0.499 + 0.866i)36-s + (−0.866 + 0.5i)37-s + ⋯
L(s)  = 1  + (−0.866 + 0.5i)2-s + (0.499 − 0.866i)4-s + i·5-s + 0.999i·8-s + (−0.5 + 0.866i)9-s + (−0.5 − 0.866i)10-s + (−0.5 − 0.866i)16-s + (−0.5 + 0.866i)17-s − 0.999i·18-s + (0.866 + 0.499i)20-s + (0.5 + 0.866i)29-s + (0.866 + 0.499i)32-s − 0.999i·34-s + (0.499 + 0.866i)36-s + (−0.866 + 0.5i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 676 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.265 - 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 676 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.265 - 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(676\)    =    \(2^{2} \cdot 13^{2}\)
Sign: $-0.265 - 0.964i$
Analytic conductor: \(0.337367\)
Root analytic conductor: \(0.580833\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{676} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 676,\ (\ :0),\ -0.265 - 0.964i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5711930066\)
\(L(\frac12)\) \(\approx\) \(0.5711930066\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 - 0.5i)T \)
13 \( 1 \)
good3 \( 1 + (0.5 - 0.866i)T^{2} \)
5 \( 1 - iT - T^{2} \)
7 \( 1 + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (-0.5 + 0.866i)T^{2} \)
17 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
19 \( 1 + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.5 - 0.866i)T^{2} \)
29 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
41 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
43 \( 1 + (0.5 + 0.866i)T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + T + T^{2} \)
59 \( 1 + (-0.5 - 0.866i)T^{2} \)
61 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + (-0.5 - 0.866i)T^{2} \)
73 \( 1 + iT - T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + (-1.73 + i)T + (0.5 - 0.866i)T^{2} \)
97 \( 1 + (-1.73 - i)T + (0.5 + 0.866i)T^{2} \)
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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

−10.74691411300486468608097107938, −10.26820207767872637651649169159, −9.103693646329961492069687257275, −8.330322232210478869438892599588, −7.50173042721762220985295565719, −6.67732483656634020697300820469, −5.88734466128779035642889046141, −4.78696513787114465596855014276, −3.08695566288795130996984523323, −1.95662763632202046635197634862, 0.852200475815374476339511689986, 2.45626233076909153801889847746, 3.69714944855823528783316673315, 4.83363210795382071037005506306, 6.12952934496471266055220023215, 7.11380780006616797039149253379, 8.162831434962240569151754772311, 8.926379145636046520401617200361, 9.380377363190463904985710672692, 10.32895554073747545570199048374

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