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Properties

Label 2-2919-2919.290-c0-0-0
Degree $2$
Conductor $2919$
Sign $0.999 - 0.0339i$
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  + (0.983 − 0.181i)3-s + (−0.613 − 0.789i)4-s + (0.419 + 0.907i)7-s + (0.934 − 0.356i)9-s + (−0.746 − 0.665i)12-s + (0.362 + 1.96i)13-s + (−0.247 + 0.968i)16-s + (0.735 − 0.402i)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 + (0.160 − 0.421i)31-s + (−0.854 − 0.519i)36-s + (1.31 + 1.40i)37-s + ⋯
L(s)  = 1  + (0.983 − 0.181i)3-s + (−0.613 − 0.789i)4-s + (0.419 + 0.907i)7-s + (0.934 − 0.356i)9-s + (−0.746 − 0.665i)12-s + (0.362 + 1.96i)13-s + (−0.247 + 0.968i)16-s + (0.735 − 0.402i)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 + (0.160 − 0.421i)31-s + (−0.854 − 0.519i)36-s + (1.31 + 1.40i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2919 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0339i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2919 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0339i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2919\)    =    \(3 \cdot 7 \cdot 139\)
Sign: $0.999 - 0.0339i$
Analytic conductor: \(1.45677\)
Root analytic conductor: \(1.20696\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2919} (290, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2919,\ (\ :0),\ 0.999 - 0.0339i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.655266714\)
\(L(\frac12)\) \(\approx\) \(1.655266714\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.983 + 0.181i)T \)
7 \( 1 + (-0.419 - 0.907i)T \)
139 \( 1 + (-0.877 - 0.480i)T \)
good2 \( 1 + (0.613 + 0.789i)T^{2} \)
5 \( 1 + (0.682 + 0.730i)T^{2} \)
11 \( 1 + (0.576 - 0.816i)T^{2} \)
13 \( 1 + (-0.362 - 1.96i)T + (-0.934 + 0.356i)T^{2} \)
17 \( 1 + (0.998 + 0.0455i)T^{2} \)
19 \( 1 + (-0.735 + 0.402i)T + (0.538 - 0.842i)T^{2} \)
23 \( 1 + (0.0227 + 0.999i)T^{2} \)
29 \( 1 + (0.247 - 0.968i)T^{2} \)
31 \( 1 + (-0.160 + 0.421i)T + (-0.746 - 0.665i)T^{2} \)
37 \( 1 + (-1.31 - 1.40i)T + (-0.0682 + 0.997i)T^{2} \)
41 \( 1 + (0.983 - 0.181i)T^{2} \)
43 \( 1 + (1.53 - 0.887i)T + (0.5 - 0.866i)T^{2} \)
47 \( 1 + (0.682 - 0.730i)T^{2} \)
53 \( 1 + (0.377 - 0.926i)T^{2} \)
59 \( 1 + (0.247 + 0.968i)T^{2} \)
61 \( 1 + (0.280 + 0.608i)T + (-0.648 + 0.761i)T^{2} \)
67 \( 1 + (-0.739 + 1.42i)T + (-0.576 - 0.816i)T^{2} \)
71 \( 1 + (0.974 - 0.225i)T^{2} \)
73 \( 1 + (0.373 + 1.79i)T + (-0.917 + 0.398i)T^{2} \)
79 \( 1 + (1.65 + 0.150i)T + (0.983 + 0.181i)T^{2} \)
83 \( 1 + (0.995 + 0.0909i)T^{2} \)
89 \( 1 + (-0.0682 - 0.997i)T^{2} \)
97 \( 1 + (-0.613 + 1.06i)T + (-0.5 - 0.866i)T^{2} \)
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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

−8.879634176816589913301865705341, −8.459942561758638531999313532357, −7.61638343879177487436129735170, −6.51376007595189101917067727020, −6.08984978748189856223579304497, −4.75307034191499515502808461449, −4.46593694419235456666885395815, −3.28222282285307966186888921831, −2.12564905354833405567337391308, −1.46683798672094664570819279503, 1.10752093756514698282035769478, 2.61523885802402902672345194965, 3.51260338394729100968076579198, 3.88968076427419797827287773115, 4.91857362827945829784015793687, 5.66854105945124295871406129688, 7.20646035703010566105381654325, 7.55205104700700929510731922394, 8.227381708308568361866955458216, 8.672003507782365206937680641439

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