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Properties

Label 2-1300-1300.687-c0-0-0
Degree $2$
Conductor $1300$
Sign $0.857 - 0.515i$
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  + (0.994 + 0.104i)2-s + (0.978 + 0.207i)4-s + (0.669 + 0.743i)5-s + (0.951 + 0.309i)8-s + (−0.994 + 0.104i)9-s + (0.587 + 0.809i)10-s + (0.104 − 0.994i)13-s + (0.913 + 0.406i)16-s + (−0.715 − 0.0375i)17-s − 0.999·18-s + (0.5 + 0.866i)20-s + (−0.104 + 0.994i)25-s + (0.207 − 0.978i)26-s + (−0.155 − 0.139i)29-s + (0.866 + 0.499i)32-s + ⋯
L(s)  = 1  + (0.994 + 0.104i)2-s + (0.978 + 0.207i)4-s + (0.669 + 0.743i)5-s + (0.951 + 0.309i)8-s + (−0.994 + 0.104i)9-s + (0.587 + 0.809i)10-s + (0.104 − 0.994i)13-s + (0.913 + 0.406i)16-s + (−0.715 − 0.0375i)17-s − 0.999·18-s + (0.5 + 0.866i)20-s + (−0.104 + 0.994i)25-s + (0.207 − 0.978i)26-s + (−0.155 − 0.139i)29-s + (0.866 + 0.499i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.857 - 0.515i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.857 - 0.515i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1300\)    =    \(2^{2} \cdot 5^{2} \cdot 13\)
Sign: $0.857 - 0.515i$
Analytic conductor: \(0.648784\)
Root analytic conductor: \(0.805471\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1300} (687, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1300,\ (\ :0),\ 0.857 - 0.515i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.058952445\)
\(L(\frac12)\) \(\approx\) \(2.058952445\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.994 - 0.104i)T \)
5 \( 1 + (-0.669 - 0.743i)T \)
13 \( 1 + (-0.104 + 0.994i)T \)
good3 \( 1 + (0.994 - 0.104i)T^{2} \)
7 \( 1 + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.207 - 0.978i)T^{2} \)
17 \( 1 + (0.715 + 0.0375i)T + (0.994 + 0.104i)T^{2} \)
19 \( 1 + (-0.994 - 0.104i)T^{2} \)
23 \( 1 + (-0.207 + 0.978i)T^{2} \)
29 \( 1 + (0.155 + 0.139i)T + (0.104 + 0.994i)T^{2} \)
31 \( 1 + (-0.587 + 0.809i)T^{2} \)
37 \( 1 + (1.81 + 0.809i)T + (0.669 + 0.743i)T^{2} \)
41 \( 1 + (-0.0375 - 0.0977i)T + (-0.743 + 0.669i)T^{2} \)
43 \( 1 + (-0.866 + 0.5i)T^{2} \)
47 \( 1 + (0.809 - 0.587i)T^{2} \)
53 \( 1 + (-0.970 - 0.494i)T + (0.587 + 0.809i)T^{2} \)
59 \( 1 + (0.207 + 0.978i)T^{2} \)
61 \( 1 + (0.379 - 0.169i)T + (0.669 - 0.743i)T^{2} \)
67 \( 1 + (0.913 - 0.406i)T^{2} \)
71 \( 1 + (0.406 - 0.913i)T^{2} \)
73 \( 1 + (-1.14 + 1.58i)T + (-0.309 - 0.951i)T^{2} \)
79 \( 1 + (-0.809 + 0.587i)T^{2} \)
83 \( 1 + (0.809 + 0.587i)T^{2} \)
89 \( 1 + (1.53 - 1.24i)T + (0.207 - 0.978i)T^{2} \)
97 \( 1 + (-0.244 + 1.14i)T + (-0.913 - 0.406i)T^{2} \)
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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

−10.26371395352124619847823900124, −9.069853602008452946158167842360, −8.146045846607251412296738981157, −7.22590037766520617204951579870, −6.43749835317191251956864454449, −5.67691599373036973438100497938, −5.09175097632931672350494409160, −3.70442482657364917117713577620, −2.90684865158929080291673313068, −2.03782338911143849716472275378, 1.63318153096880559041715893442, 2.61158158359485515112304356229, 3.81061762976280261875552634794, 4.77658997267888032848688645054, 5.47369940322445866107631529577, 6.30386559793607306338076888386, 6.97457575369804227600984923305, 8.276682047708071837143560177088, 8.945416100863837271163041935756, 9.811242829667858731950746657031

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