Properties

Label 2-2013-2013.731-c0-0-0
Degree $2$
Conductor $2013$
Sign $0.993 - 0.111i$
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  + (−0.550 − 1.69i)2-s + (0.809 + 0.587i)3-s + (−1.76 + 1.27i)4-s + (0.550 − 1.69i)6-s + (1.69 + 1.23i)8-s + (0.309 + 0.951i)9-s + (−0.987 − 0.156i)11-s − 2.17·12-s + (0.587 + 1.80i)13-s + (0.481 − 1.48i)16-s + (−0.610 + 1.87i)17-s + (1.44 − 1.04i)18-s + (−0.951 − 0.690i)19-s + (0.278 + 1.76i)22-s + 0.907·23-s + (0.647 + 1.99i)24-s + ⋯
L(s)  = 1  + (−0.550 − 1.69i)2-s + (0.809 + 0.587i)3-s + (−1.76 + 1.27i)4-s + (0.550 − 1.69i)6-s + (1.69 + 1.23i)8-s + (0.309 + 0.951i)9-s + (−0.987 − 0.156i)11-s − 2.17·12-s + (0.587 + 1.80i)13-s + (0.481 − 1.48i)16-s + (−0.610 + 1.87i)17-s + (1.44 − 1.04i)18-s + (−0.951 − 0.690i)19-s + (0.278 + 1.76i)22-s + 0.907·23-s + (0.647 + 1.99i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 - 0.111i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 - 0.111i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2013\)    =    \(3 \cdot 11 \cdot 61\)
Sign: $0.993 - 0.111i$
Analytic conductor: \(1.00461\)
Root analytic conductor: \(1.00230\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2013} (731, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2013,\ (\ :0),\ 0.993 - 0.111i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8131426719\)
\(L(\frac12)\) \(\approx\) \(0.8131426719\)
\(L(1)\) 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 \( 1 + (0.987 + 0.156i)T \)
61 \( 1 + (0.309 - 0.951i)T \)
good2 \( 1 + (0.550 + 1.69i)T + (-0.809 + 0.587i)T^{2} \)
5 \( 1 + (0.809 + 0.587i)T^{2} \)
7 \( 1 + (-0.309 + 0.951i)T^{2} \)
13 \( 1 + (-0.587 - 1.80i)T + (-0.809 + 0.587i)T^{2} \)
17 \( 1 + (0.610 - 1.87i)T + (-0.809 - 0.587i)T^{2} \)
19 \( 1 + (0.951 + 0.690i)T + (0.309 + 0.951i)T^{2} \)
23 \( 1 - 0.907T + T^{2} \)
29 \( 1 + (0.734 - 0.533i)T + (0.309 - 0.951i)T^{2} \)
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 \( 1 + (-0.550 - 1.69i)T + (-0.809 + 0.587i)T^{2} \)
59 \( 1 + (0.253 - 0.183i)T + (0.309 - 0.951i)T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + (-0.437 + 1.34i)T + (-0.809 - 0.587i)T^{2} \)
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 \( 1 - 1.97T + T^{2} \)
97 \( 1 + (-0.809 + 0.587i)T^{2} \)
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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

−9.336929154832688404564314597204, −8.751820783514707359801544574318, −8.467903935721223449065730720121, −7.39311678535344379720476929835, −6.15954234466438213923559779687, −4.66662071898105129600216741018, −4.14103014724390735943653857330, −3.40005783515136628396722255583, −2.27177855063693084980982456094, −1.79171537658583323397369183110, 0.62717365881658090889106592400, 2.40853350610774955147545431173, 3.53107258319266281330594764090, 4.88996722618729370298071563825, 5.57840257805418243254082078867, 6.38245418203761754687778292771, 7.25774315447076356739393037218, 7.75475622602943396681157217545, 8.242114833971634052657487660891, 9.009320017199039480735817569662

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