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Properties

Label 2-1300-260.159-c0-0-1
Degree $2$
Conductor $1300$
Sign $0.322 + 0.946i$
Analytic cond. $0.648784$
Root an. cond. $0.805471$
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 − 0.999i·8-s + (0.5 − 0.866i)9-s + (−0.866 + 0.5i)13-s + (−0.5 − 0.866i)16-s + (0.866 + 0.5i)17-s − 0.999i·18-s + (−0.499 + 0.866i)26-s + (−0.5 − 0.866i)29-s + (−0.866 − 0.499i)32-s + 0.999·34-s + (−0.499 − 0.866i)36-s + (−0.866 + 0.5i)37-s + (0.5 + 0.866i)41-s + ⋯
L(s)  = 1  + (0.866 − 0.5i)2-s + (0.499 − 0.866i)4-s − 0.999i·8-s + (0.5 − 0.866i)9-s + (−0.866 + 0.5i)13-s + (−0.5 − 0.866i)16-s + (0.866 + 0.5i)17-s − 0.999i·18-s + (−0.499 + 0.866i)26-s + (−0.5 − 0.866i)29-s + (−0.866 − 0.499i)32-s + 0.999·34-s + (−0.499 − 0.866i)36-s + (−0.866 + 0.5i)37-s + (0.5 + 0.866i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.322 + 0.946i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.322 + 0.946i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1300\)    =    \(2^{2} \cdot 5^{2} \cdot 13\)
Sign: $0.322 + 0.946i$
Analytic conductor: \(0.648784\)
Root analytic conductor: \(0.805471\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1300} (1199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1300,\ (\ :0),\ 0.322 + 0.946i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.760722933\)
\(L(\frac12)\) \(\approx\) \(1.760722933\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 + 0.5i)T \)
5 \( 1 \)
13 \( 1 + (0.866 - 0.5i)T \)
good3 \( 1 + (-0.5 + 0.866i)T^{2} \)
7 \( 1 + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.5 - 0.866i)T^{2} \)
17 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
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 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
43 \( 1 + (-0.5 - 0.866i)T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 - iT - T^{2} \)
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 \( 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2} \)
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

−9.758210179183789447318335778353, −9.268346365327046581355003035838, −7.909674559126548668581743745213, −7.01152129392295245060938114223, −6.27274976130521990472467734300, −5.39969822103896721978137736743, −4.39971536108601574372988807332, −3.67801980432094684887005176503, −2.59556630440025032927538730155, −1.33705424340649811963945238265, 2.01168699708265087898790573557, 3.07910241370270908332979874371, 4.10854964338987685375928354527, 5.18012736352962858740059839442, 5.49208539868665828003532021453, 6.86090680238316886857376080204, 7.41291184211611895805915827409, 8.068292750052940453585784080375, 9.079882062600352825233161105369, 10.16316791636623927342116823540

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