Properties

Label 2-2013-2013.1280-c0-0-1
Degree $2$
Conductor $2013$
Sign $0.991 + 0.132i$
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  + (−1.59 − 1.16i)2-s + (−0.309 + 0.951i)3-s + (0.896 + 2.76i)4-s + (1.59 − 1.16i)6-s + (1.16 − 3.57i)8-s + (−0.809 − 0.587i)9-s + (−0.453 + 0.891i)11-s − 2.90·12-s + (0.951 + 0.690i)13-s + (−3.65 + 2.65i)16-s + (0.734 − 0.533i)17-s + (0.610 + 1.87i)18-s + (0.587 − 1.80i)19-s + (1.76 − 0.896i)22-s + 0.312·23-s + (3.03 + 2.20i)24-s + ⋯
L(s)  = 1  + (−1.59 − 1.16i)2-s + (−0.309 + 0.951i)3-s + (0.896 + 2.76i)4-s + (1.59 − 1.16i)6-s + (1.16 − 3.57i)8-s + (−0.809 − 0.587i)9-s + (−0.453 + 0.891i)11-s − 2.90·12-s + (0.951 + 0.690i)13-s + (−3.65 + 2.65i)16-s + (0.734 − 0.533i)17-s + (0.610 + 1.87i)18-s + (0.587 − 1.80i)19-s + (1.76 − 0.896i)22-s + 0.312·23-s + (3.03 + 2.20i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 + 0.132i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 + 0.132i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2013\)    =    \(3 \cdot 11 \cdot 61\)
Sign: $0.991 + 0.132i$
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: \((2,\ 2013,\ (\ :0),\ 0.991 + 0.132i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4852535592\)
\(L(\frac12)\) \(\approx\) \(0.4852535592\)
\(L(1)\) 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 \( 1 + (0.453 - 0.891i)T \)
61 \( 1 + (-0.809 + 0.587i)T \)
good2 \( 1 + (1.59 + 1.16i)T + (0.309 + 0.951i)T^{2} \)
5 \( 1 + (-0.309 + 0.951i)T^{2} \)
7 \( 1 + (0.809 - 0.587i)T^{2} \)
13 \( 1 + (-0.951 - 0.690i)T + (0.309 + 0.951i)T^{2} \)
17 \( 1 + (-0.734 + 0.533i)T + (0.309 - 0.951i)T^{2} \)
19 \( 1 + (-0.587 + 1.80i)T + (-0.809 - 0.587i)T^{2} \)
23 \( 1 - 0.312T + T^{2} \)
29 \( 1 + (-0.0966 - 0.297i)T + (-0.809 + 0.587i)T^{2} \)
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 \( 1 + (-1.59 - 1.16i)T + (0.309 + 0.951i)T^{2} \)
59 \( 1 + (0.550 + 1.69i)T + (-0.809 + 0.587i)T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + (1.14 - 0.831i)T + (0.309 - 0.951i)T^{2} \)
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 \( 1 - 0.907T + T^{2} \)
97 \( 1 + (0.309 + 0.951i)T^{2} \)
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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

−9.361545522635880362145810526402, −8.970135157740281171812690987934, −8.158350349838644512508943122793, −7.21557220939545658635586182526, −6.53582368575083875791290771893, −4.98768370274604267830291410010, −4.15131215957184382979810895111, −3.16574436270994045511738618163, −2.41462667961955188958378324491, −0.929316804963404035556181063398, 0.908207317474104170195334532246, 1.73252143870362796964371001492, 3.23989939922879071389592503232, 5.33467587050083111257618190156, 5.76352124991637515693168357185, 6.27135349087650206820881067608, 7.28974762968105947401783170338, 7.86164762769791967031086848557, 8.352248882090240702450501967735, 8.950332347479972230745548710203

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