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Properties

Label 2-2919-2919.26-c0-0-0
Degree $2$
Conductor $2919$
Sign $0.164 - 0.986i$
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  + (0.898 + 0.439i)3-s + (0.648 + 0.761i)4-s + (−0.291 + 0.956i)7-s + (0.613 + 0.789i)9-s + (0.247 + 0.968i)12-s + (0.585 − 1.19i)13-s + (−0.158 + 0.987i)16-s + (−0.553 + 0.182i)19-s + (−0.682 + 0.730i)21-s + (−0.460 − 0.887i)25-s + (0.203 + 0.979i)27-s + (−0.917 + 0.398i)28-s + (1.33 + 1.03i)31-s + (−0.203 + 0.979i)36-s + (−0.713 − 1.37i)37-s + ⋯
L(s)  = 1  + (0.898 + 0.439i)3-s + (0.648 + 0.761i)4-s + (−0.291 + 0.956i)7-s + (0.613 + 0.789i)9-s + (0.247 + 0.968i)12-s + (0.585 − 1.19i)13-s + (−0.158 + 0.987i)16-s + (−0.553 + 0.182i)19-s + (−0.682 + 0.730i)21-s + (−0.460 − 0.887i)25-s + (0.203 + 0.979i)27-s + (−0.917 + 0.398i)28-s + (1.33 + 1.03i)31-s + (−0.203 + 0.979i)36-s + (−0.713 − 1.37i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2919 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.164 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2919 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.164 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2919\)    =    \(3 \cdot 7 \cdot 139\)
Sign: $0.164 - 0.986i$
Analytic conductor: \(1.45677\)
Root analytic conductor: \(1.20696\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2919} (26, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2919,\ (\ :0),\ 0.164 - 0.986i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.937230596\)
\(L(\frac12)\) \(\approx\) \(1.937230596\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.898 - 0.439i)T \)
7 \( 1 + (0.291 - 0.956i)T \)
139 \( 1 + (-0.949 - 0.313i)T \)
good2 \( 1 + (-0.648 - 0.761i)T^{2} \)
5 \( 1 + (0.460 + 0.887i)T^{2} \)
11 \( 1 + (-0.682 - 0.730i)T^{2} \)
13 \( 1 + (-0.585 + 1.19i)T + (-0.613 - 0.789i)T^{2} \)
17 \( 1 + (-0.113 - 0.993i)T^{2} \)
19 \( 1 + (0.553 - 0.182i)T + (0.803 - 0.595i)T^{2} \)
23 \( 1 + (0.746 - 0.665i)T^{2} \)
29 \( 1 + (0.158 - 0.987i)T^{2} \)
31 \( 1 + (-1.33 - 1.03i)T + (0.247 + 0.968i)T^{2} \)
37 \( 1 + (0.713 + 1.37i)T + (-0.576 + 0.816i)T^{2} \)
41 \( 1 + (0.898 + 0.439i)T^{2} \)
43 \( 1 + (0.690 + 0.398i)T + (0.5 + 0.866i)T^{2} \)
47 \( 1 + (0.460 - 0.887i)T^{2} \)
53 \( 1 + (0.983 - 0.181i)T^{2} \)
59 \( 1 + (0.158 + 0.987i)T^{2} \)
61 \( 1 + (-0.0398 + 0.130i)T + (-0.829 - 0.557i)T^{2} \)
67 \( 1 + (0.0417 - 0.0181i)T + (0.682 - 0.730i)T^{2} \)
71 \( 1 + (-0.538 + 0.842i)T^{2} \)
73 \( 1 + (-1.64 + 0.461i)T + (0.854 - 0.519i)T^{2} \)
79 \( 1 + (1.94 - 0.449i)T + (0.898 - 0.439i)T^{2} \)
83 \( 1 + (-0.974 + 0.225i)T^{2} \)
89 \( 1 + (-0.576 - 0.816i)T^{2} \)
97 \( 1 + (0.648 + 1.12i)T + (-0.5 + 0.866i)T^{2} \)
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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

−8.773525812623175164609856942438, −8.434225878611288763929338702855, −7.87761785947473399280255200751, −6.93090444529921859269802412125, −6.13553669835019479204528280600, −5.26232673097444442077829284570, −4.15199579250357033908739488092, −3.32709049734019476475579480255, −2.73190657035051470273240175769, −1.89358884205660012633306726113, 1.16671652727372791108199118672, 1.96119126870822889107133981353, 3.01511663151272647472264151797, 3.94092638946537325699130199531, 4.72923193350704892798577358713, 6.07348921563206047566682439334, 6.64430580185603215253757432227, 7.12012749110292534066491239891, 7.975380106189471931388883673576, 8.741098041702302102851028965904

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