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Properties

Label 2-2013-2013.548-c0-0-1
Degree $2$
Conductor $2013$
Sign $-0.781 + 0.624i$
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  + (−0.587 + 1.80i)2-s + (−0.809 + 0.587i)3-s + (−2.11 − 1.53i)4-s + (−0.587 − 1.80i)6-s + (2.48 − 1.80i)8-s + (0.309 − 0.951i)9-s + (0.587 − 0.809i)11-s + 2.61·12-s + (−0.190 + 0.587i)13-s + (1.00 + 3.07i)16-s + (0.363 + 1.11i)17-s + (1.53 + 1.11i)18-s + (−1.30 + 0.951i)19-s + (1.11 + 1.53i)22-s + 1.90·23-s + (−0.951 + 2.92i)24-s + ⋯
L(s)  = 1  + (−0.587 + 1.80i)2-s + (−0.809 + 0.587i)3-s + (−2.11 − 1.53i)4-s + (−0.587 − 1.80i)6-s + (2.48 − 1.80i)8-s + (0.309 − 0.951i)9-s + (0.587 − 0.809i)11-s + 2.61·12-s + (−0.190 + 0.587i)13-s + (1.00 + 3.07i)16-s + (0.363 + 1.11i)17-s + (1.53 + 1.11i)18-s + (−1.30 + 0.951i)19-s + (1.11 + 1.53i)22-s + 1.90·23-s + (−0.951 + 2.92i)24-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(2013\)    =    \(3 \cdot 11 \cdot 61\)
Sign: $-0.781 + 0.624i$
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: \((2,\ 2013,\ (\ :0),\ -0.781 + 0.624i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4311711201\)
\(L(\frac12)\) \(\approx\) \(0.4311711201\)
\(L(1)\) 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 \( 1 + (-0.587 + 0.809i)T \)
61 \( 1 + (-0.309 - 0.951i)T \)
good2 \( 1 + (0.587 - 1.80i)T + (-0.809 - 0.587i)T^{2} \)
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 \( 1 + (-0.363 - 1.11i)T + (-0.809 + 0.587i)T^{2} \)
19 \( 1 + (1.30 - 0.951i)T + (0.309 - 0.951i)T^{2} \)
23 \( 1 - 1.90T + T^{2} \)
29 \( 1 + (1.53 + 1.11i)T + (0.309 + 0.951i)T^{2} \)
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 \( 1 + (0.587 - 1.80i)T + (-0.809 - 0.587i)T^{2} \)
59 \( 1 + (-0.951 - 0.690i)T + (0.309 + 0.951i)T^{2} \)
67 \( 1 - T^{2} \)
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 \( 1 - 1.17T + T^{2} \)
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

−9.321364745169415949367662439387, −9.130129100362571053384273773412, −8.189538163681598818638451121026, −7.37949256744599393116198831818, −6.52660828911821298850655096656, −5.94783403026096639825794861741, −5.49337356752383248371788280581, −4.33570247108385539387851417761, −3.83030563793163751999054737396, −1.29081119819759120988045586093, 0.48443209250786288639840106179, 1.72190055060124084475343864170, 2.52686301325488560054297447323, 3.62865318046311392893220522857, 4.74368398463025571244444858850, 5.22205232692650113714869190565, 6.78946668605101264266745874177, 7.37290855386548632066359462917, 8.361397196930968427410527632238, 9.146379218180762329153438656580

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