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Properties

Label 2-2940-35.9-c1-0-35
Degree $2$
Conductor $2940$
Sign $-0.208 + 0.978i$
Analytic cond. $23.4760$
Root an. cond. $4.84520$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.866 − 0.5i)3-s + (1.86 − 1.23i)5-s + (0.499 − 0.866i)9-s + (−2 − 3.46i)11-s − 2i·13-s + (1 − 2i)15-s + (1.73 − i)17-s + (1 − 1.73i)19-s + (5.19 + 3i)23-s + (1.96 − 4.59i)25-s − 0.999i·27-s − 6·29-s + (3 + 5.19i)31-s + (−3.46 − 1.99i)33-s + (−3.46 − 2i)37-s + ⋯
L(s)  = 1  + (0.499 − 0.288i)3-s + (0.834 − 0.550i)5-s + (0.166 − 0.288i)9-s + (−0.603 − 1.04i)11-s − 0.554i·13-s + (0.258 − 0.516i)15-s + (0.420 − 0.242i)17-s + (0.229 − 0.397i)19-s + (1.08 + 0.625i)23-s + (0.392 − 0.919i)25-s − 0.192i·27-s − 1.11·29-s + (0.538 + 0.933i)31-s + (−0.603 − 0.348i)33-s + (−0.569 − 0.328i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2940 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.208 + 0.978i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2940 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.208 + 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2940\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 7^{2}\)
Sign: $-0.208 + 0.978i$
Analytic conductor: \(23.4760\)
Root analytic conductor: \(4.84520\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2940} (1549, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2940,\ (\ :1/2),\ -0.208 + 0.978i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.362928339\)
\(L(\frac12)\) \(\approx\) \(2.362928339\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-0.866 + 0.5i)T \)
5 \( 1 + (-1.86 + 1.23i)T \)
7 \( 1 \)
good11 \( 1 + (2 + 3.46i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 2iT - 13T^{2} \)
17 \( 1 + (-1.73 + i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1 + 1.73i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-5.19 - 3i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 6T + 29T^{2} \)
31 \( 1 + (-3 - 5.19i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (3.46 + 2i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 + 4iT - 43T^{2} \)
47 \( 1 + (3.46 + 2i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.73 - i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (2 + 3.46i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1 - 1.73i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-10.3 + 6i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 8T + 71T^{2} \)
73 \( 1 + (12.1 - 7i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-8 + 13.8i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 16iT - 83T^{2} \)
89 \( 1 + (8 - 13.8i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 14iT - 97T^{2} \)
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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

−8.707763570719145750707176869144, −7.83573526978306044238244309575, −7.14792000810988804289552473047, −6.15933166796120161760579674337, −5.43335278125285412818325819804, −4.88639726427674749393869043368, −3.46186318704185183757578909696, −2.87665743589810890999832571309, −1.74168765203007639876791434728, −0.68110247243081575114457671525, 1.56588118527570995651870460858, 2.40816305410420984106293619747, 3.21025064371054842045046101526, 4.27815002342273676576949134078, 5.08335084409772773506573169675, 5.89654474489274338284294546164, 6.81410752259115899589736618328, 7.40957935936991970023926136576, 8.215585377433439200775261724115, 9.138234362477815312467753953376

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