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Properties

Label 2-2919-2919.1046-c0-0-0
Degree $2$
Conductor $2919$
Sign $0.749 - 0.661i$
Analytic cond. $1.45677$
Root an. cond. $1.20696$
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  + (−0.648 + 0.761i)3-s + (−0.377 − 0.926i)4-s + (−0.995 + 0.0909i)7-s + (−0.158 − 0.987i)9-s + (0.949 + 0.313i)12-s + (−0.730 − 0.622i)13-s + (−0.715 + 0.699i)16-s + (0.0453 + 1.99i)19-s + (0.576 − 0.816i)21-s + (−0.682 + 0.730i)25-s + (0.854 + 0.519i)27-s + (0.460 + 0.887i)28-s + (1.88 + 0.303i)31-s + (−0.854 + 0.519i)36-s + (1.31 − 1.40i)37-s + ⋯
L(s)  = 1  + (−0.648 + 0.761i)3-s + (−0.377 − 0.926i)4-s + (−0.995 + 0.0909i)7-s + (−0.158 − 0.987i)9-s + (0.949 + 0.313i)12-s + (−0.730 − 0.622i)13-s + (−0.715 + 0.699i)16-s + (0.0453 + 1.99i)19-s + (0.576 − 0.816i)21-s + (−0.682 + 0.730i)25-s + (0.854 + 0.519i)27-s + (0.460 + 0.887i)28-s + (1.88 + 0.303i)31-s + (−0.854 + 0.519i)36-s + (1.31 − 1.40i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2919 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.749 - 0.661i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2919 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.749 - 0.661i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2919\)    =    \(3 \cdot 7 \cdot 139\)
Sign: $0.749 - 0.661i$
Analytic conductor: \(1.45677\)
Root analytic conductor: \(1.20696\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2919} (1046, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2919,\ (\ :0),\ 0.749 - 0.661i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6350484454\)
\(L(\frac12)\) \(\approx\) \(0.6350484454\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.648 - 0.761i)T \)
7 \( 1 + (0.995 - 0.0909i)T \)
139 \( 1 + (0.0227 - 0.999i)T \)
good2 \( 1 + (0.377 + 0.926i)T^{2} \)
5 \( 1 + (0.682 - 0.730i)T^{2} \)
11 \( 1 + (0.576 + 0.816i)T^{2} \)
13 \( 1 + (0.730 + 0.622i)T + (0.158 + 0.987i)T^{2} \)
17 \( 1 + (-0.538 - 0.842i)T^{2} \)
19 \( 1 + (-0.0453 - 1.99i)T + (-0.998 + 0.0455i)T^{2} \)
23 \( 1 + (-0.877 + 0.480i)T^{2} \)
29 \( 1 + (0.715 - 0.699i)T^{2} \)
31 \( 1 + (-1.88 - 0.303i)T + (0.949 + 0.313i)T^{2} \)
37 \( 1 + (-1.31 + 1.40i)T + (-0.0682 - 0.997i)T^{2} \)
41 \( 1 + (-0.648 + 0.761i)T^{2} \)
43 \( 1 + (-1.53 + 0.887i)T + (0.5 - 0.866i)T^{2} \)
47 \( 1 + (0.682 + 0.730i)T^{2} \)
53 \( 1 + (0.613 - 0.789i)T^{2} \)
59 \( 1 + (0.715 + 0.699i)T^{2} \)
61 \( 1 + (-0.666 + 0.0609i)T + (0.983 - 0.181i)T^{2} \)
67 \( 1 + (-0.104 - 0.201i)T + (-0.576 + 0.816i)T^{2} \)
71 \( 1 + (-0.291 - 0.956i)T^{2} \)
73 \( 1 + (0.373 - 1.79i)T + (-0.917 - 0.398i)T^{2} \)
79 \( 1 + (0.752 - 1.63i)T + (-0.648 - 0.761i)T^{2} \)
83 \( 1 + (-0.419 + 0.907i)T^{2} \)
89 \( 1 + (-0.0682 + 0.997i)T^{2} \)
97 \( 1 + (-0.377 + 0.653i)T + (-0.5 - 0.866i)T^{2} \)
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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

−9.369246793341978845117035448799, −8.473029704604500610371721379405, −7.43675154268918430077582511699, −6.43748035283100249147869583279, −5.74929054731916830597116764391, −5.46252024702515201310240741514, −4.30490116005309003354849314259, −3.71966981886729060357197248049, −2.51230992312095141046567701727, −0.905555977044906304395745240979, 0.59038543842823875180523720794, 2.46018592670713524439383948826, 2.93529099543431116004823359899, 4.42593910091603961832312895428, 4.71900174735774557479975358265, 6.10360185082563175632759837070, 6.60469245790685119520011586523, 7.32773329404629594623903504272, 7.918620378337068137122842057841, 8.819476124000221454141271638947

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