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Properties

Label 2-2919-2919.1655-c0-0-0
Degree $2$
Conductor $2919$
Sign $-0.478 + 0.877i$
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  + (0.538 − 0.842i)3-s + (−0.291 + 0.956i)4-s + (−0.0227 − 0.999i)7-s + (−0.419 − 0.907i)9-s + (0.648 + 0.761i)12-s + (−1.33 − 0.851i)13-s + (−0.829 − 0.557i)16-s + (0.0171 − 0.0421i)19-s + (−0.854 − 0.519i)21-s + (−0.203 − 0.979i)25-s + (−0.990 − 0.136i)27-s + (0.962 + 0.269i)28-s + (−0.568 − 0.262i)31-s + (0.990 − 0.136i)36-s + (−0.0277 − 0.133i)37-s + ⋯
L(s)  = 1  + (0.538 − 0.842i)3-s + (−0.291 + 0.956i)4-s + (−0.0227 − 0.999i)7-s + (−0.419 − 0.907i)9-s + (0.648 + 0.761i)12-s + (−1.33 − 0.851i)13-s + (−0.829 − 0.557i)16-s + (0.0171 − 0.0421i)19-s + (−0.854 − 0.519i)21-s + (−0.203 − 0.979i)25-s + (−0.990 − 0.136i)27-s + (0.962 + 0.269i)28-s + (−0.568 − 0.262i)31-s + (0.990 − 0.136i)36-s + (−0.0277 − 0.133i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2919 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.478 + 0.877i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2919 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.478 + 0.877i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2919\)    =    \(3 \cdot 7 \cdot 139\)
Sign: $-0.478 + 0.877i$
Analytic conductor: \(1.45677\)
Root analytic conductor: \(1.20696\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2919} (1655, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2919,\ (\ :0),\ -0.478 + 0.877i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9595513863\)
\(L(\frac12)\) \(\approx\) \(0.9595513863\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.538 + 0.842i)T \)
7 \( 1 + (0.0227 + 0.999i)T \)
139 \( 1 + (0.377 + 0.926i)T \)
good2 \( 1 + (0.291 - 0.956i)T^{2} \)
5 \( 1 + (0.203 + 0.979i)T^{2} \)
11 \( 1 + (-0.854 + 0.519i)T^{2} \)
13 \( 1 + (1.33 + 0.851i)T + (0.419 + 0.907i)T^{2} \)
17 \( 1 + (0.247 - 0.968i)T^{2} \)
19 \( 1 + (-0.0171 + 0.0421i)T + (-0.715 - 0.699i)T^{2} \)
23 \( 1 + (0.613 - 0.789i)T^{2} \)
29 \( 1 + (0.829 + 0.557i)T^{2} \)
31 \( 1 + (0.568 + 0.262i)T + (0.648 + 0.761i)T^{2} \)
37 \( 1 + (0.0277 + 0.133i)T + (-0.917 + 0.398i)T^{2} \)
41 \( 1 + (0.538 - 0.842i)T^{2} \)
43 \( 1 + (-0.467 + 0.269i)T + (0.5 - 0.866i)T^{2} \)
47 \( 1 + (0.203 - 0.979i)T^{2} \)
53 \( 1 + (-0.974 + 0.225i)T^{2} \)
59 \( 1 + (0.829 - 0.557i)T^{2} \)
61 \( 1 + (0.0209 + 0.919i)T + (-0.998 + 0.0455i)T^{2} \)
67 \( 1 + (-1.79 - 0.504i)T + (0.854 + 0.519i)T^{2} \)
71 \( 1 + (0.949 + 0.313i)T^{2} \)
73 \( 1 + (-0.519 - 1.46i)T + (-0.775 + 0.631i)T^{2} \)
79 \( 1 + (0.199 + 0.109i)T + (0.538 + 0.842i)T^{2} \)
83 \( 1 + (-0.877 - 0.480i)T^{2} \)
89 \( 1 + (-0.917 - 0.398i)T^{2} \)
97 \( 1 + (-0.291 + 0.505i)T + (-0.5 - 0.866i)T^{2} \)
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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

−8.292483645262359803877236640488, −8.039495970633868956040476657607, −7.17419316391690397691098353124, −6.91716178500678844062068423356, −5.67584360749945175227768719340, −4.60346364425475428670507736313, −3.79716265412375529266928546901, −2.97828152818537023069780629386, −2.16147901282995125068616276742, −0.52937006372962523775227793742, 1.84021803826878485408316244723, 2.55515687160221454039934570919, 3.70675101289865362615641601453, 4.72493445998388126178259585248, 5.16961195419188948992112068382, 5.91194143561624737300191162342, 6.89679709563929238389568715866, 7.83058408112207353119703079069, 8.763710529368524672499281449646, 9.359259124343684815530670960524

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