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Properties

Label 2-2919-2919.1349-c0-0-0
Degree $2$
Conductor $2919$
Sign $0.231 - 0.972i$
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  + (−0.877 + 0.480i)3-s + (−0.803 + 0.595i)4-s + (0.715 + 0.699i)7-s + (0.538 − 0.842i)9-s + (0.419 − 0.907i)12-s + (0.422 + 0.771i)13-s + (0.291 − 0.956i)16-s + (1.18 − 0.797i)19-s + (−0.962 − 0.269i)21-s + (0.775 + 0.631i)25-s + (−0.0682 + 0.997i)27-s + (−0.990 − 0.136i)28-s + (1.66 − 1.06i)31-s + (0.0682 + 0.997i)36-s + (−1.05 − 0.861i)37-s + ⋯
L(s)  = 1  + (−0.877 + 0.480i)3-s + (−0.803 + 0.595i)4-s + (0.715 + 0.699i)7-s + (0.538 − 0.842i)9-s + (0.419 − 0.907i)12-s + (0.422 + 0.771i)13-s + (0.291 − 0.956i)16-s + (1.18 − 0.797i)19-s + (−0.962 − 0.269i)21-s + (0.775 + 0.631i)25-s + (−0.0682 + 0.997i)27-s + (−0.990 − 0.136i)28-s + (1.66 − 1.06i)31-s + (0.0682 + 0.997i)36-s + (−1.05 − 0.861i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2919 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.231 - 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2919 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.231 - 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2919\)    =    \(3 \cdot 7 \cdot 139\)
Sign: $0.231 - 0.972i$
Analytic conductor: \(1.45677\)
Root analytic conductor: \(1.20696\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2919} (1349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2919,\ (\ :0),\ 0.231 - 0.972i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8772114196\)
\(L(\frac12)\) \(\approx\) \(0.8772114196\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.877 - 0.480i)T \)
7 \( 1 + (-0.715 - 0.699i)T \)
139 \( 1 + (-0.829 - 0.557i)T \)
good2 \( 1 + (0.803 - 0.595i)T^{2} \)
5 \( 1 + (-0.775 - 0.631i)T^{2} \)
11 \( 1 + (-0.962 + 0.269i)T^{2} \)
13 \( 1 + (-0.422 - 0.771i)T + (-0.538 + 0.842i)T^{2} \)
17 \( 1 + (-0.613 - 0.789i)T^{2} \)
19 \( 1 + (-1.18 + 0.797i)T + (0.377 - 0.926i)T^{2} \)
23 \( 1 + (0.898 - 0.439i)T^{2} \)
29 \( 1 + (-0.291 + 0.956i)T^{2} \)
31 \( 1 + (-1.66 + 1.06i)T + (0.419 - 0.907i)T^{2} \)
37 \( 1 + (1.05 + 0.861i)T + (0.203 + 0.979i)T^{2} \)
41 \( 1 + (-0.877 + 0.480i)T^{2} \)
43 \( 1 + (-0.235 - 0.136i)T + (0.5 + 0.866i)T^{2} \)
47 \( 1 + (-0.775 + 0.631i)T^{2} \)
53 \( 1 + (0.113 + 0.993i)T^{2} \)
59 \( 1 + (-0.291 - 0.956i)T^{2} \)
61 \( 1 + (-1.22 - 1.19i)T + (0.0227 + 0.999i)T^{2} \)
67 \( 1 + (1.94 + 0.267i)T + (0.962 + 0.269i)T^{2} \)
71 \( 1 + (0.158 - 0.987i)T^{2} \)
73 \( 1 + (-0.386 + 0.547i)T + (-0.334 - 0.942i)T^{2} \)
79 \( 1 + (0.369 - 1.44i)T + (-0.877 - 0.480i)T^{2} \)
83 \( 1 + (-0.247 + 0.968i)T^{2} \)
89 \( 1 + (0.203 - 0.979i)T^{2} \)
97 \( 1 + (-0.803 - 1.39i)T + (-0.5 + 0.866i)T^{2} \)
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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

−9.131232489944323539600241246812, −8.550148353866664423972964188673, −7.59817847550523275146812286970, −6.84544292980786899505460801410, −5.82535038690798411007910121982, −5.12816804338566388009029518201, −4.55875246744660142367993955277, −3.76555889201585633625624393521, −2.70106723848308426991038443611, −1.11734543310774889077582151158, 0.864811621755801853315622021476, 1.55596316756136369641566682834, 3.23113269052217205648206100047, 4.38285018308165025960178915466, 4.98783430114628469145297938220, 5.60399824722143107722373325658, 6.42717816068218123877700952453, 7.22209722741862540516159464960, 8.132473217001299726344107025848, 8.515706511243460254747916901735

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