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Properties

Label 2-2013-2013.1280-c0-0-5
Degree $2$
Conductor $2013$
Sign $-0.132 + 0.991i$
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  + (0.253 + 0.183i)2-s + (−0.309 + 0.951i)3-s + (−0.278 − 0.857i)4-s + (−0.253 + 0.183i)6-s + (0.183 − 0.565i)8-s + (−0.809 − 0.587i)9-s + (−0.891 − 0.453i)11-s + 0.902·12-s + (−0.951 − 0.690i)13-s + (−0.579 + 0.420i)16-s + (1.44 − 1.04i)17-s + (−0.0966 − 0.297i)18-s + (−0.587 + 1.80i)19-s + (−0.142 − 0.278i)22-s − 1.97·23-s + (0.481 + 0.349i)24-s + ⋯
L(s)  = 1  + (0.253 + 0.183i)2-s + (−0.309 + 0.951i)3-s + (−0.278 − 0.857i)4-s + (−0.253 + 0.183i)6-s + (0.183 − 0.565i)8-s + (−0.809 − 0.587i)9-s + (−0.891 − 0.453i)11-s + 0.902·12-s + (−0.951 − 0.690i)13-s + (−0.579 + 0.420i)16-s + (1.44 − 1.04i)17-s + (−0.0966 − 0.297i)18-s + (−0.587 + 1.80i)19-s + (−0.142 − 0.278i)22-s − 1.97·23-s + (0.481 + 0.349i)24-s + ⋯

Functional equation

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

Invariants

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

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5608247001\)
\(L(\frac12)\) \(\approx\) \(0.5608247001\)
\(L(1)\) 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 \( 1 + (0.891 + 0.453i)T \)
61 \( 1 + (-0.809 + 0.587i)T \)
good2 \( 1 + (-0.253 - 0.183i)T + (0.309 + 0.951i)T^{2} \)
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 \( 1 + (-1.44 + 1.04i)T + (0.309 - 0.951i)T^{2} \)
19 \( 1 + (0.587 - 1.80i)T + (-0.809 - 0.587i)T^{2} \)
23 \( 1 + 1.97T + T^{2} \)
29 \( 1 + (0.610 + 1.87i)T + (-0.809 + 0.587i)T^{2} \)
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 \( 1 + (0.253 + 0.183i)T + (0.309 + 0.951i)T^{2} \)
59 \( 1 + (0.280 + 0.863i)T + (-0.809 + 0.587i)T^{2} \)
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 \( 1 - 1.78T + T^{2} \)
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

−9.634690254087784660444519417295, −8.109973456091292815633538881791, −7.903122162554543105072211791849, −6.33150873089600231272982928448, −5.73760733263985400279337137353, −5.25441193249403093615286350110, −4.35458572093999033544042982656, −3.51669647537966434429883008385, −2.28857873144811104849140819493, −0.36115825192076849760638840886, 1.83560640296761417309634171290, 2.65132010394315456074986714331, 3.68585334913103043298345001490, 4.85927894234284236681512391420, 5.38541126563518939731804711603, 6.58461960890962295189290422030, 7.34684611172296055902072129242, 7.81365917174095555209131472160, 8.586445694238827572734485777460, 9.425363360489763649037932417714

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