Properties

Degree $2$
Conductor $2013$
Sign $-0.624 - 0.781i$
Motivic weight $0$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.190 + 0.587i)2-s + (−0.809 + 0.587i)3-s + (0.5 + 0.363i)4-s + (−0.190 − 0.587i)6-s + (−0.809 + 0.587i)8-s + (0.309 − 0.951i)9-s + (0.809 + 0.587i)11-s − 0.618·12-s + (0.190 − 0.587i)13-s + (0.5 + 1.53i)17-s + (0.5 + 0.363i)18-s + (1.30 − 0.951i)19-s + (−0.5 + 0.363i)22-s − 0.618·23-s + (0.309 − 0.951i)24-s + (−0.809 + 0.587i)25-s + ⋯
L(s)  = 1  + (−0.190 + 0.587i)2-s + (−0.809 + 0.587i)3-s + (0.5 + 0.363i)4-s + (−0.190 − 0.587i)6-s + (−0.809 + 0.587i)8-s + (0.309 − 0.951i)9-s + (0.809 + 0.587i)11-s − 0.618·12-s + (0.190 − 0.587i)13-s + (0.5 + 1.53i)17-s + (0.5 + 0.363i)18-s + (1.30 − 0.951i)19-s + (−0.5 + 0.363i)22-s − 0.618·23-s + (0.309 − 0.951i)24-s + (−0.809 + 0.587i)25-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(2013\)    =    \(3 \cdot 11 \cdot 61\)
Sign: $-0.624 - 0.781i$
Motivic weight: \(0\)
Character: $\chi_{2013} (548, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2013,\ (\ :0),\ -0.624 - 0.781i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9463058085\)
\(L(\frac12)\) \(\approx\) \(0.9463058085\)
\(L(1)\) 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 \( 1 + (-0.809 - 0.587i)T \)
61 \( 1 + (-0.309 - 0.951i)T \)
good2 \( 1 + (0.190 - 0.587i)T + (-0.809 - 0.587i)T^{2} \)
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 \( 1 + (-0.5 - 1.53i)T + (-0.809 + 0.587i)T^{2} \)
19 \( 1 + (-1.30 + 0.951i)T + (0.309 - 0.951i)T^{2} \)
23 \( 1 + 0.618T + T^{2} \)
29 \( 1 + (-0.5 - 0.363i)T + (0.309 + 0.951i)T^{2} \)
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 \( 1 + (0.190 - 0.587i)T + (-0.809 - 0.587i)T^{2} \)
59 \( 1 + (1.30 + 0.951i)T + (0.309 + 0.951i)T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + (0.618 + 1.90i)T + (-0.809 + 0.587i)T^{2} \)
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 \( 1 - 1.61T + T^{2} \)
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

−9.520612680525360702048693309579, −8.912768938518411169833433777530, −7.85609979949378841327558817601, −7.28251599020888648555186589317, −6.29807843740301208693657585439, −5.91190183912080268775618973618, −4.95478659404725486318513384440, −3.88778399802942666935190567251, −3.12173711911705180404061157220, −1.51593202432189572704968351758, 0.874909967690796563013590971992, 1.80312335816194354538799742484, 2.93198356080364066099228272194, 4.02536232319546232996318940945, 5.30296073741477493261327587951, 5.93556959148486658659893375617, 6.64392642561724848515497545755, 7.34860568183918830171399597611, 8.204050235311925216634685313506, 9.409013791707354756600063934988

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