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Properties

Label 2-2013-2013.1280-c0-0-2
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

\(L(\frac{1}{2})\) \(\approx\) \(2.513158698\)
\(L(\frac12)\) \(\approx\) \(2.513158698\)
\(L(1)\) 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

−9.402144043521016448398045747251, −8.662612579260402951067978459269, −8.172050986530125953477521175782, −6.77838207716986966401116782257, −6.46156787261306576356159457435, −5.69831436418721362017734805671, −4.84991091218968970928822563513, −4.20202242231903346616152222797, −3.53479963713586264170083191315, −2.61963555680888732834256276915, 1.28312408580997660719761067881, 1.95066699871069439480579623908, 3.11103694408735472848410184554, 3.85346582297929288369709311902, 4.95040944027768742673482957607, 5.59799984362840312300822206759, 6.33535204415830178707116721469, 6.98427572104612079928940480706, 8.008854437140394227201877642288, 9.304399859833851683134181243605

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