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Properties

Label 2-2919-2919.269-c0-0-0
Degree $2$
Conductor $2919$
Sign $0.738 - 0.674i$
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  + (0.995 + 0.0909i)3-s + (−0.898 + 0.439i)4-s + (−0.538 − 0.842i)7-s + (0.983 + 0.181i)9-s + (−0.934 + 0.356i)12-s + (−0.127 + 1.39i)13-s + (0.613 − 0.789i)16-s + (−0.267 + 1.04i)19-s + (−0.460 − 0.887i)21-s + (0.917 − 0.398i)25-s + (0.962 + 0.269i)27-s + (0.854 + 0.519i)28-s + (0.359 + 1.95i)31-s + (−0.962 + 0.269i)36-s + (1.81 − 0.789i)37-s + ⋯
L(s)  = 1  + (0.995 + 0.0909i)3-s + (−0.898 + 0.439i)4-s + (−0.538 − 0.842i)7-s + (0.983 + 0.181i)9-s + (−0.934 + 0.356i)12-s + (−0.127 + 1.39i)13-s + (0.613 − 0.789i)16-s + (−0.267 + 1.04i)19-s + (−0.460 − 0.887i)21-s + (0.917 − 0.398i)25-s + (0.962 + 0.269i)27-s + (0.854 + 0.519i)28-s + (0.359 + 1.95i)31-s + (−0.962 + 0.269i)36-s + (1.81 − 0.789i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2919 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.738 - 0.674i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2919 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.738 - 0.674i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2919\)    =    \(3 \cdot 7 \cdot 139\)
Sign: $0.738 - 0.674i$
Analytic conductor: \(1.45677\)
Root analytic conductor: \(1.20696\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2919} (269, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2919,\ (\ :0),\ 0.738 - 0.674i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.363199468\)
\(L(\frac12)\) \(\approx\) \(1.363199468\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.995 - 0.0909i)T \)
7 \( 1 + (0.538 + 0.842i)T \)
139 \( 1 + (-0.247 - 0.968i)T \)
good2 \( 1 + (0.898 - 0.439i)T^{2} \)
5 \( 1 + (-0.917 + 0.398i)T^{2} \)
11 \( 1 + (-0.460 + 0.887i)T^{2} \)
13 \( 1 + (0.127 - 1.39i)T + (-0.983 - 0.181i)T^{2} \)
17 \( 1 + (-0.0227 - 0.999i)T^{2} \)
19 \( 1 + (0.267 - 1.04i)T + (-0.877 - 0.480i)T^{2} \)
23 \( 1 + (-0.715 + 0.699i)T^{2} \)
29 \( 1 + (-0.613 + 0.789i)T^{2} \)
31 \( 1 + (-0.359 - 1.95i)T + (-0.934 + 0.356i)T^{2} \)
37 \( 1 + (-1.81 + 0.789i)T + (0.682 - 0.730i)T^{2} \)
41 \( 1 + (0.995 + 0.0909i)T^{2} \)
43 \( 1 + (-0.899 + 0.519i)T + (0.5 - 0.866i)T^{2} \)
47 \( 1 + (-0.917 - 0.398i)T^{2} \)
53 \( 1 + (-0.829 - 0.557i)T^{2} \)
59 \( 1 + (-0.613 - 0.789i)T^{2} \)
61 \( 1 + (-0.621 - 0.971i)T + (-0.419 + 0.907i)T^{2} \)
67 \( 1 + (1.62 + 0.986i)T + (0.460 + 0.887i)T^{2} \)
71 \( 1 + (-0.113 + 0.993i)T^{2} \)
73 \( 1 + (0.315 - 0.256i)T + (0.203 - 0.979i)T^{2} \)
79 \( 1 + (0.582 - 0.0265i)T + (0.995 - 0.0909i)T^{2} \)
83 \( 1 + (-0.998 + 0.0455i)T^{2} \)
89 \( 1 + (0.682 + 0.730i)T^{2} \)
97 \( 1 + (-0.898 + 1.55i)T + (-0.5 - 0.866i)T^{2} \)
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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

−8.912665002272415982476051062947, −8.493588869365611512535582708474, −7.52326036915132543006712119038, −7.07222435241469316638634211587, −6.10749720146097000175813458691, −4.71254571344579078084214188411, −4.23905088401243662650620079208, −3.55998539339244869077358064959, −2.67614217331328520931127730286, −1.31042618490261582309275802480, 0.906210015624039427179721692482, 2.47794068435146511528267004891, 3.03734235318054937923020751548, 4.12253690780719593585770955638, 4.89903947334347827209777336323, 5.78589206977558129287938925990, 6.49515333247678730046275034696, 7.65943755351307351653484195985, 8.180207382262613419314169398634, 8.976990116812727503377595116672

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