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Properties

Label 2-1300-1300.259-c0-0-1
Degree $2$
Conductor $1300$
Sign $0.876 - 0.481i$
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  + (0.809 + 0.587i)2-s + (0.309 + 0.951i)4-s + (0.309 − 0.951i)5-s + (−0.309 + 0.951i)8-s + (0.809 − 0.587i)9-s + (0.809 − 0.587i)10-s + (−0.809 − 0.587i)13-s + (−0.809 + 0.587i)16-s + (1.11 + 0.363i)17-s + 18-s + 20-s + (−0.809 − 0.587i)25-s + (−0.309 − 0.951i)26-s + (0.5 + 1.53i)29-s − 32-s + ⋯
L(s)  = 1  + (0.809 + 0.587i)2-s + (0.309 + 0.951i)4-s + (0.309 − 0.951i)5-s + (−0.309 + 0.951i)8-s + (0.809 − 0.587i)9-s + (0.809 − 0.587i)10-s + (−0.809 − 0.587i)13-s + (−0.809 + 0.587i)16-s + (1.11 + 0.363i)17-s + 18-s + 20-s + (−0.809 − 0.587i)25-s + (−0.309 − 0.951i)26-s + (0.5 + 1.53i)29-s − 32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.876 - 0.481i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.876 - 0.481i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1300\)    =    \(2^{2} \cdot 5^{2} \cdot 13\)
Sign: $0.876 - 0.481i$
Analytic conductor: \(0.648784\)
Root analytic conductor: \(0.805471\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1300} (259, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1300,\ (\ :0),\ 0.876 - 0.481i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.790421094\)
\(L(\frac12)\) \(\approx\) \(1.790421094\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.809 - 0.587i)T \)
5 \( 1 + (-0.309 + 0.951i)T \)
13 \( 1 + (0.809 + 0.587i)T \)
good3 \( 1 + (-0.809 + 0.587i)T^{2} \)
7 \( 1 - T^{2} \)
11 \( 1 + (0.309 + 0.951i)T^{2} \)
17 \( 1 + (-1.11 - 0.363i)T + (0.809 + 0.587i)T^{2} \)
19 \( 1 + (-0.809 - 0.587i)T^{2} \)
23 \( 1 + (0.309 + 0.951i)T^{2} \)
29 \( 1 + (-0.5 - 1.53i)T + (-0.809 + 0.587i)T^{2} \)
31 \( 1 + (-0.809 - 0.587i)T^{2} \)
37 \( 1 + (1.30 - 0.951i)T + (0.309 - 0.951i)T^{2} \)
41 \( 1 + (1.11 + 1.53i)T + (-0.309 + 0.951i)T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + (0.809 - 0.587i)T^{2} \)
53 \( 1 + (1.80 - 0.587i)T + (0.809 - 0.587i)T^{2} \)
59 \( 1 + (0.309 - 0.951i)T^{2} \)
61 \( 1 + (-0.5 - 0.363i)T + (0.309 + 0.951i)T^{2} \)
67 \( 1 + (0.809 + 0.587i)T^{2} \)
71 \( 1 + (-0.809 + 0.587i)T^{2} \)
73 \( 1 + (-0.5 - 0.363i)T + (0.309 + 0.951i)T^{2} \)
79 \( 1 + (0.809 - 0.587i)T^{2} \)
83 \( 1 + (0.809 + 0.587i)T^{2} \)
89 \( 1 + (0.690 - 0.951i)T + (-0.309 - 0.951i)T^{2} \)
97 \( 1 + (0.5 + 1.53i)T + (-0.809 + 0.587i)T^{2} \)
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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

−9.872396290749837580804657018672, −8.927772236197571003558152687121, −8.206556341923945529503545202196, −7.32635018122234770411954621216, −6.60507829949639870985158534854, −5.48338217199191172364336052898, −5.04737195388509812474018239632, −4.02274640985583990502053933316, −3.09617288005201587909402820999, −1.56100148856249326080737018526, 1.72492685825827688896841110592, 2.62239382316628216093577842622, 3.62417435187499690843348748564, 4.63121880377543023509445274656, 5.43817507661809815025805033895, 6.45388441945623399627814550131, 7.11771675487351229396692243274, 7.925130732869435691170053980334, 9.516378359738968531118578904220, 9.928439649682449476173985238708

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