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Properties

Label 2-2919-2919.1307-c0-0-0
Degree $2$
Conductor $2919$
Sign $0.564 - 0.825i$
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  + (−0.247 + 0.968i)3-s + (0.949 + 0.313i)4-s + (−0.377 − 0.926i)7-s + (−0.877 − 0.480i)9-s + (−0.538 + 0.842i)12-s + (0.437 + 0.111i)13-s + (0.803 + 0.595i)16-s + (0.220 − 0.722i)19-s + (0.990 − 0.136i)21-s + (0.334 + 0.942i)25-s + (0.682 − 0.730i)27-s + (−0.0682 − 0.997i)28-s + (0.730 + 1.33i)31-s + (−0.682 − 0.730i)36-s + (0.614 + 1.72i)37-s + ⋯
L(s)  = 1  + (−0.247 + 0.968i)3-s + (0.949 + 0.313i)4-s + (−0.377 − 0.926i)7-s + (−0.877 − 0.480i)9-s + (−0.538 + 0.842i)12-s + (0.437 + 0.111i)13-s + (0.803 + 0.595i)16-s + (0.220 − 0.722i)19-s + (0.990 − 0.136i)21-s + (0.334 + 0.942i)25-s + (0.682 − 0.730i)27-s + (−0.0682 − 0.997i)28-s + (0.730 + 1.33i)31-s + (−0.682 − 0.730i)36-s + (0.614 + 1.72i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2919 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.564 - 0.825i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2919 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.564 - 0.825i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2919\)    =    \(3 \cdot 7 \cdot 139\)
Sign: $0.564 - 0.825i$
Analytic conductor: \(1.45677\)
Root analytic conductor: \(1.20696\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2919} (1307, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2919,\ (\ :0),\ 0.564 - 0.825i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.394337454\)
\(L(\frac12)\) \(\approx\) \(1.394337454\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.247 - 0.968i)T \)
7 \( 1 + (0.377 + 0.926i)T \)
139 \( 1 + (0.291 + 0.956i)T \)
good2 \( 1 + (-0.949 - 0.313i)T^{2} \)
5 \( 1 + (-0.334 - 0.942i)T^{2} \)
11 \( 1 + (0.990 + 0.136i)T^{2} \)
13 \( 1 + (-0.437 - 0.111i)T + (0.877 + 0.480i)T^{2} \)
17 \( 1 + (-0.898 + 0.439i)T^{2} \)
19 \( 1 + (-0.220 + 0.722i)T + (-0.829 - 0.557i)T^{2} \)
23 \( 1 + (-0.974 - 0.225i)T^{2} \)
29 \( 1 + (-0.803 - 0.595i)T^{2} \)
31 \( 1 + (-0.730 - 1.33i)T + (-0.538 + 0.842i)T^{2} \)
37 \( 1 + (-0.614 - 1.72i)T + (-0.775 + 0.631i)T^{2} \)
41 \( 1 + (-0.247 + 0.968i)T^{2} \)
43 \( 1 + (-1.72 - 0.997i)T + (0.5 + 0.866i)T^{2} \)
47 \( 1 + (-0.334 + 0.942i)T^{2} \)
53 \( 1 + (0.746 - 0.665i)T^{2} \)
59 \( 1 + (-0.803 + 0.595i)T^{2} \)
61 \( 1 + (0.726 + 1.78i)T + (-0.715 + 0.699i)T^{2} \)
67 \( 1 + (0.135 + 1.98i)T + (-0.990 + 0.136i)T^{2} \)
71 \( 1 + (0.648 - 0.761i)T^{2} \)
73 \( 1 + (0.530 - 1.02i)T + (-0.576 - 0.816i)T^{2} \)
79 \( 1 + (-1.14 + 1.47i)T + (-0.247 - 0.968i)T^{2} \)
83 \( 1 + (0.613 - 0.789i)T^{2} \)
89 \( 1 + (-0.775 - 0.631i)T^{2} \)
97 \( 1 + (0.949 + 1.64i)T + (-0.5 + 0.866i)T^{2} \)
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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

−9.219719697681135744748672205128, −8.270277003331720161903598065391, −7.52677137349185775502334738208, −6.61755925659720430849614706807, −6.23226841711687329854451053515, −5.08241529887470327376426382109, −4.33368422690684114727579766603, −3.34347314834947767512842400848, −2.89976772532120265955059683070, −1.26233954818257528487277277597, 1.04679019675163306646228716523, 2.30913335124062909306949365826, 2.66332770229039867267076647234, 3.99161479288563611356378141601, 5.48439634778940325450885290815, 5.87606601506423330383925418515, 6.42922235384660702645328844667, 7.29428883882911034021772285582, 7.905356780668472196776713328050, 8.693695645582900572588160866730

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