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Properties

Label 2-2013-2013.1280-c0-0-4
Degree $2$
Conductor $2013$
Sign $0.794 - 0.606i$
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  + (1.30 + 0.951i)2-s + (0.309 − 0.951i)3-s + (0.500 + 1.53i)4-s + (1.30 − 0.951i)6-s + (−0.309 + 0.951i)8-s + (−0.809 − 0.587i)9-s + (0.309 + 0.951i)11-s + 1.61·12-s + (1.30 + 0.951i)13-s + (−0.5 + 0.363i)17-s + (−0.499 − 1.53i)18-s + (0.190 − 0.587i)19-s + (−0.499 + 1.53i)22-s − 1.61·23-s + (0.809 + 0.587i)24-s + (0.309 − 0.951i)25-s + ⋯
L(s)  = 1  + (1.30 + 0.951i)2-s + (0.309 − 0.951i)3-s + (0.500 + 1.53i)4-s + (1.30 − 0.951i)6-s + (−0.309 + 0.951i)8-s + (−0.809 − 0.587i)9-s + (0.309 + 0.951i)11-s + 1.61·12-s + (1.30 + 0.951i)13-s + (−0.5 + 0.363i)17-s + (−0.499 − 1.53i)18-s + (0.190 − 0.587i)19-s + (−0.499 + 1.53i)22-s − 1.61·23-s + (0.809 + 0.587i)24-s + (0.309 − 0.951i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.794 - 0.606i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.794 - 0.606i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2013\)    =    \(3 \cdot 11 \cdot 61\)
Sign: $0.794 - 0.606i$
Analytic conductor: \(1.00461\)
Root analytic conductor: \(1.00230\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2013} (1280, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2013,\ (\ :0),\ 0.794 - 0.606i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.595023628\)
\(L(\frac12)\) \(\approx\) \(2.595023628\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.309 + 0.951i)T \)
11 \( 1 + (-0.309 - 0.951i)T \)
61 \( 1 + (0.809 - 0.587i)T \)
good2 \( 1 + (-1.30 - 0.951i)T + (0.309 + 0.951i)T^{2} \)
5 \( 1 + (-0.309 + 0.951i)T^{2} \)
7 \( 1 + (0.809 - 0.587i)T^{2} \)
13 \( 1 + (-1.30 - 0.951i)T + (0.309 + 0.951i)T^{2} \)
17 \( 1 + (0.5 - 0.363i)T + (0.309 - 0.951i)T^{2} \)
19 \( 1 + (-0.190 + 0.587i)T + (-0.809 - 0.587i)T^{2} \)
23 \( 1 + 1.61T + T^{2} \)
29 \( 1 + (0.5 + 1.53i)T + (-0.809 + 0.587i)T^{2} \)
31 \( 1 + (-0.309 - 0.951i)T^{2} \)
37 \( 1 + (0.809 - 0.587i)T^{2} \)
41 \( 1 + (0.809 + 0.587i)T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (0.809 + 0.587i)T^{2} \)
53 \( 1 + (-1.30 - 0.951i)T + (0.309 + 0.951i)T^{2} \)
59 \( 1 + (-0.190 - 0.587i)T + (-0.809 + 0.587i)T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + (1.61 - 1.17i)T + (0.309 - 0.951i)T^{2} \)
73 \( 1 + (0.5 + 1.53i)T + (-0.809 + 0.587i)T^{2} \)
79 \( 1 + (-0.309 - 0.951i)T^{2} \)
83 \( 1 + (-0.309 + 0.951i)T^{2} \)
89 \( 1 - 0.618T + T^{2} \)
97 \( 1 + (1.61 + 1.17i)T + (0.309 + 0.951i)T^{2} \)
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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

−9.081414514982793929138229788939, −8.326935421552818523811158676441, −7.56393890845154947448000449248, −6.88230029723289910654450813392, −6.20159755150888266028503701119, −5.81820661742036333616827116408, −4.29358346710920206862928083509, −4.11513672882863889336920846109, −2.75485978529463267966960097434, −1.68886661510674863768926731397, 1.57544670175381383041692720923, 2.87667747755197645752423453914, 3.59221123046221468534299891148, 3.95289697488769425724936961715, 5.19702476317934490884801871754, 5.59971976181835431385930481035, 6.43063132910694996140041416001, 7.965353268169358114555959370119, 8.591145763642572841227068996806, 9.441747903614451860815079359378

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