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Properties

Label 2-2013-2013.548-c0-0-6
Degree $2$
Conductor $2013$
Sign $-0.111 + 0.993i$
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  + (0.280 − 0.863i)2-s + (0.809 − 0.587i)3-s + (0.142 + 0.103i)4-s + (−0.280 − 0.863i)6-s + (0.863 − 0.627i)8-s + (0.309 − 0.951i)9-s + (−0.156 − 0.987i)11-s + 0.175·12-s + (−0.587 + 1.80i)13-s + (−0.245 − 0.754i)16-s + (−0.0966 − 0.297i)17-s + (−0.734 − 0.533i)18-s + (0.951 − 0.690i)19-s + (−0.896 − 0.142i)22-s − 1.78·23-s + (0.329 − 1.01i)24-s + ⋯
L(s)  = 1  + (0.280 − 0.863i)2-s + (0.809 − 0.587i)3-s + (0.142 + 0.103i)4-s + (−0.280 − 0.863i)6-s + (0.863 − 0.627i)8-s + (0.309 − 0.951i)9-s + (−0.156 − 0.987i)11-s + 0.175·12-s + (−0.587 + 1.80i)13-s + (−0.245 − 0.754i)16-s + (−0.0966 − 0.297i)17-s + (−0.734 − 0.533i)18-s + (0.951 − 0.690i)19-s + (−0.896 − 0.142i)22-s − 1.78·23-s + (0.329 − 1.01i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.111 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.111 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2013\)    =    \(3 \cdot 11 \cdot 61\)
Sign: $-0.111 + 0.993i$
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: \((2,\ 2013,\ (\ :0),\ -0.111 + 0.993i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.973746685\)
\(L(\frac12)\) \(\approx\) \(1.973746685\)
\(L(1)\) 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 \( 1 + (0.156 + 0.987i)T \)
61 \( 1 + (0.309 + 0.951i)T \)
good2 \( 1 + (-0.280 + 0.863i)T + (-0.809 - 0.587i)T^{2} \)
5 \( 1 + (0.809 - 0.587i)T^{2} \)
7 \( 1 + (-0.309 - 0.951i)T^{2} \)
13 \( 1 + (0.587 - 1.80i)T + (-0.809 - 0.587i)T^{2} \)
17 \( 1 + (0.0966 + 0.297i)T + (-0.809 + 0.587i)T^{2} \)
19 \( 1 + (-0.951 + 0.690i)T + (0.309 - 0.951i)T^{2} \)
23 \( 1 + 1.78T + T^{2} \)
29 \( 1 + (-1.44 - 1.04i)T + (0.309 + 0.951i)T^{2} \)
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 \( 1 + (0.280 - 0.863i)T + (-0.809 - 0.587i)T^{2} \)
59 \( 1 + (1.59 + 1.16i)T + (0.309 + 0.951i)T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + (-0.437 - 1.34i)T + (-0.809 + 0.587i)T^{2} \)
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 \( 1 - 0.312T + T^{2} \)
97 \( 1 + (-0.809 - 0.587i)T^{2} \)
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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

−9.328148373511666908211752933347, −8.332617131418161363996339568066, −7.57990015616129710962243623203, −6.89985681867617254678519576040, −6.17126076733783284938319601037, −4.74933234901712569633682457661, −3.89287871634308325799112717404, −3.07799162872453317372829016406, −2.28072648475148568048610636940, −1.36990127099163908495953238378, 1.90944607994317792711097544702, 2.78587822154144241262968422982, 3.98488651809761178148977560567, 4.79600183465085778588667269837, 5.55890626943441611992356026859, 6.29633973044382484404303095215, 7.58399240190565120753828346897, 7.77869773230945783535904697822, 8.400241055977691846586656068900, 9.816879764116882481900618892093

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