Properties

Label 2-2013-2013.548-c0-0-4
Degree $2$
Conductor $2013$
Sign $0.624 + 0.781i$
Analytic cond. $1.00461$
Root an. cond. $1.00230$
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  + (0.190 − 0.587i)2-s + (−0.809 + 0.587i)3-s + (0.5 + 0.363i)4-s + (0.190 + 0.587i)6-s + (0.809 − 0.587i)8-s + (0.309 − 0.951i)9-s + (−0.809 − 0.587i)11-s − 0.618·12-s + (0.190 − 0.587i)13-s + (−0.5 − 1.53i)17-s + (−0.5 − 0.363i)18-s + (1.30 − 0.951i)19-s + (−0.5 + 0.363i)22-s + 0.618·23-s + (−0.309 + 0.951i)24-s + (−0.809 + 0.587i)25-s + ⋯
L(s)  = 1  + (0.190 − 0.587i)2-s + (−0.809 + 0.587i)3-s + (0.5 + 0.363i)4-s + (0.190 + 0.587i)6-s + (0.809 − 0.587i)8-s + (0.309 − 0.951i)9-s + (−0.809 − 0.587i)11-s − 0.618·12-s + (0.190 − 0.587i)13-s + (−0.5 − 1.53i)17-s + (−0.5 − 0.363i)18-s + (1.30 − 0.951i)19-s + (−0.5 + 0.363i)22-s + 0.618·23-s + (−0.309 + 0.951i)24-s + (−0.809 + 0.587i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.624 + 0.781i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.624 + 0.781i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2013\)    =    \(3 \cdot 11 \cdot 61\)
Sign: $0.624 + 0.781i$
Analytic conductor: \(1.00461\)
Root analytic conductor: \(1.00230\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2013} (548, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2013,\ (\ :0),\ 0.624 + 0.781i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.136097591\)
\(L(\frac12)\) \(\approx\) \(1.136097591\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.809 - 0.587i)T \)
11 \( 1 + (0.809 + 0.587i)T \)
61 \( 1 + (-0.309 - 0.951i)T \)
good2 \( 1 + (-0.190 + 0.587i)T + (-0.809 - 0.587i)T^{2} \)
5 \( 1 + (0.809 - 0.587i)T^{2} \)
7 \( 1 + (-0.309 - 0.951i)T^{2} \)
13 \( 1 + (-0.190 + 0.587i)T + (-0.809 - 0.587i)T^{2} \)
17 \( 1 + (0.5 + 1.53i)T + (-0.809 + 0.587i)T^{2} \)
19 \( 1 + (-1.30 + 0.951i)T + (0.309 - 0.951i)T^{2} \)
23 \( 1 - 0.618T + T^{2} \)
29 \( 1 + (0.5 + 0.363i)T + (0.309 + 0.951i)T^{2} \)
31 \( 1 + (0.809 + 0.587i)T^{2} \)
37 \( 1 + (-0.309 - 0.951i)T^{2} \)
41 \( 1 + (-0.309 + 0.951i)T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (-0.309 + 0.951i)T^{2} \)
53 \( 1 + (-0.190 + 0.587i)T + (-0.809 - 0.587i)T^{2} \)
59 \( 1 + (-1.30 - 0.951i)T + (0.309 + 0.951i)T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + (-0.618 - 1.90i)T + (-0.809 + 0.587i)T^{2} \)
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 \( 1 + 1.61T + T^{2} \)
97 \( 1 + (-0.618 + 1.90i)T + (-0.809 - 0.587i)T^{2} \)
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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

−9.533973482195947095773430384360, −8.565380925853737839181462616412, −7.33913931956396647031770622194, −7.09366521541492215140423604863, −5.78026527202102135929968583138, −5.22468041487568057483296394102, −4.28606516916897942383683105997, −3.24588558521075082405944708887, −2.64410846358512875662186522678, −0.902967577569815416861286500863, 1.51168696704987709529697247973, 2.25098831899809015604863774115, 3.89692965906837451583909724345, 5.01299527421013749686078410533, 5.56919691033676277124904450063, 6.33566622316266720514617648492, 6.94408335931463795429198706644, 7.74093007752102663656858409026, 8.229076582719228173938044452713, 9.602465803014763586778516472643

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