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Properties

Label 2-2919-2919.710-c0-0-0
Degree $2$
Conductor $2919$
Sign $-0.902 - 0.430i$
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  + (−0.949 + 0.313i)3-s + (−0.0227 + 0.999i)4-s + (−0.934 + 0.356i)7-s + (0.803 − 0.595i)9-s + (−0.291 − 0.956i)12-s + (0.568 + 1.72i)13-s + (−0.998 − 0.0455i)16-s + (1.86 − 0.169i)19-s + (0.775 − 0.631i)21-s + (0.990 − 0.136i)25-s + (−0.576 + 0.816i)27-s + (−0.334 − 0.942i)28-s + (−1.10 + 1.48i)31-s + (0.576 + 0.816i)36-s + (−0.911 + 0.125i)37-s + ⋯
L(s)  = 1  + (−0.949 + 0.313i)3-s + (−0.0227 + 0.999i)4-s + (−0.934 + 0.356i)7-s + (0.803 − 0.595i)9-s + (−0.291 − 0.956i)12-s + (0.568 + 1.72i)13-s + (−0.998 − 0.0455i)16-s + (1.86 − 0.169i)19-s + (0.775 − 0.631i)21-s + (0.990 − 0.136i)25-s + (−0.576 + 0.816i)27-s + (−0.334 − 0.942i)28-s + (−1.10 + 1.48i)31-s + (0.576 + 0.816i)36-s + (−0.911 + 0.125i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2919 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.902 - 0.430i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2919 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.902 - 0.430i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2919\)    =    \(3 \cdot 7 \cdot 139\)
Sign: $-0.902 - 0.430i$
Analytic conductor: \(1.45677\)
Root analytic conductor: \(1.20696\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2919} (710, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2919,\ (\ :0),\ -0.902 - 0.430i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6511139961\)
\(L(\frac12)\) \(\approx\) \(0.6511139961\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.949 - 0.313i)T \)
7 \( 1 + (0.934 - 0.356i)T \)
139 \( 1 + (0.995 + 0.0909i)T \)
good2 \( 1 + (0.0227 - 0.999i)T^{2} \)
5 \( 1 + (-0.990 + 0.136i)T^{2} \)
11 \( 1 + (0.775 + 0.631i)T^{2} \)
13 \( 1 + (-0.568 - 1.72i)T + (-0.803 + 0.595i)T^{2} \)
17 \( 1 + (0.648 + 0.761i)T^{2} \)
19 \( 1 + (-1.86 + 0.169i)T + (0.983 - 0.181i)T^{2} \)
23 \( 1 + (-0.419 - 0.907i)T^{2} \)
29 \( 1 + (0.998 + 0.0455i)T^{2} \)
31 \( 1 + (1.10 - 1.48i)T + (-0.291 - 0.956i)T^{2} \)
37 \( 1 + (0.911 - 0.125i)T + (0.962 - 0.269i)T^{2} \)
41 \( 1 + (-0.949 + 0.313i)T^{2} \)
43 \( 1 + (1.63 - 0.942i)T + (0.5 - 0.866i)T^{2} \)
47 \( 1 + (-0.990 - 0.136i)T^{2} \)
53 \( 1 + (-0.877 + 0.480i)T^{2} \)
59 \( 1 + (0.998 - 0.0455i)T^{2} \)
61 \( 1 + (0.380 - 0.144i)T + (0.746 - 0.665i)T^{2} \)
67 \( 1 + (0.601 + 1.69i)T + (-0.775 + 0.631i)T^{2} \)
71 \( 1 + (-0.377 + 0.926i)T^{2} \)
73 \( 1 + (0.0931 + 0.0997i)T + (-0.0682 + 0.997i)T^{2} \)
79 \( 1 + (-0.0786 + 0.489i)T + (-0.949 - 0.313i)T^{2} \)
83 \( 1 + (-0.158 + 0.987i)T^{2} \)
89 \( 1 + (0.962 + 0.269i)T^{2} \)
97 \( 1 + (-0.0227 + 0.0394i)T + (-0.5 - 0.866i)T^{2} \)
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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

−9.148621939176114586766406831443, −8.826935268758515371587139883876, −7.55032291029016935962200499531, −6.76257670115591987353212430853, −6.52893043127944994270683131977, −5.32262456697890061962479226613, −4.65740643846637848265043236881, −3.60924497393262284443237412105, −3.14201493040483596613546801640, −1.57616068569019835802372650363, 0.50037721008425096615553724933, 1.43388737316130789838161236058, 2.93028686398105439278378999769, 3.87989950762946986577602972772, 5.20842496765478565151397250380, 5.49584325281471901827352857568, 6.19196775829261110524292405141, 7.04157352992251906600717460144, 7.56167289632648540826249480816, 8.689071393848002340903095843315

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