Properties

Degree 2
Conductor $ 3 \cdot 11 \cdot 61 $
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

\( d \)  =  \(2\)
\( N \)  =  \(2013\)    =    \(3 \cdot 11 \cdot 61\)
\( \varepsilon \)  =  $0.624 - 0.781i$
motivic weight  =  \(0\)
character  :  $\chi_{2013} (731, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 2013,\ (\ :0),\ 0.624 - 0.781i)\)
\(L(\frac{1}{2})\)  \(\approx\)  \(1.136097591\)
\(L(\frac12)\)  \(\approx\)  \(1.136097591\)
\(L(1)\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;11,\;61\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{3,\;11,\;61\}$, then $F_p(T)$ is a polynomial of degree at most 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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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−9.602465803014763586778516472643, −8.229076582719228173938044452713, −7.74093007752102663656858409026, −6.94408335931463795429198706644, −6.33566622316266720514617648492, −5.56919691033676277124904450063, −5.01299527421013749686078410533, −3.89692965906837451583909724345, −2.25098831899809015604863774115, −1.51168696704987709529697247973, 0.902967577569815416861286500863, 2.64410846358512875662186522678, 3.24588558521075082405944708887, 4.28606516916897942383683105997, 5.22468041487568057483296394102, 5.78026527202102135929968583138, 7.09366521541492215140423604863, 7.33913931956396647031770622194, 8.565380925853737839181462616412, 9.533973482195947095773430384360

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