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Properties

Label 2-3040-760.739-c0-0-0
Degree $2$
Conductor $3040$
Sign $0.776 - 0.630i$
Analytic cond. $1.51715$
Root an. cond. $1.23172$
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  + (−0.939 + 0.342i)5-s + (−0.939 − 1.62i)7-s + (0.173 + 0.984i)9-s + (0.173 − 0.300i)11-s + (−0.766 + 0.642i)13-s + (−0.173 + 0.984i)19-s + (1.43 + 0.524i)23-s + (0.766 − 0.642i)25-s + (1.43 + 1.20i)35-s + 1.53·37-s + (0.266 + 0.223i)41-s + (−0.5 − 0.866i)45-s + (0.173 + 0.984i)47-s + (−1.26 + 2.19i)49-s + (1.76 + 0.642i)53-s + ⋯
L(s)  = 1  + (−0.939 + 0.342i)5-s + (−0.939 − 1.62i)7-s + (0.173 + 0.984i)9-s + (0.173 − 0.300i)11-s + (−0.766 + 0.642i)13-s + (−0.173 + 0.984i)19-s + (1.43 + 0.524i)23-s + (0.766 − 0.642i)25-s + (1.43 + 1.20i)35-s + 1.53·37-s + (0.266 + 0.223i)41-s + (−0.5 − 0.866i)45-s + (0.173 + 0.984i)47-s + (−1.26 + 2.19i)49-s + (1.76 + 0.642i)53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.776 - 0.630i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.776 - 0.630i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3040\)    =    \(2^{5} \cdot 5 \cdot 19\)
Sign: $0.776 - 0.630i$
Analytic conductor: \(1.51715\)
Root analytic conductor: \(1.23172\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3040} (2639, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3040,\ (\ :0),\ 0.776 - 0.630i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8283632252\)
\(L(\frac12)\) \(\approx\) \(0.8283632252\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (0.939 - 0.342i)T \)
19 \( 1 + (0.173 - 0.984i)T \)
good3 \( 1 + (-0.173 - 0.984i)T^{2} \)
7 \( 1 + (0.939 + 1.62i)T + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (-0.173 + 0.300i)T + (-0.5 - 0.866i)T^{2} \)
13 \( 1 + (0.766 - 0.642i)T + (0.173 - 0.984i)T^{2} \)
17 \( 1 + (0.939 + 0.342i)T^{2} \)
23 \( 1 + (-1.43 - 0.524i)T + (0.766 + 0.642i)T^{2} \)
29 \( 1 + (0.939 - 0.342i)T^{2} \)
31 \( 1 + (0.5 - 0.866i)T^{2} \)
37 \( 1 - 1.53T + T^{2} \)
41 \( 1 + (-0.266 - 0.223i)T + (0.173 + 0.984i)T^{2} \)
43 \( 1 + (-0.766 + 0.642i)T^{2} \)
47 \( 1 + (-0.173 - 0.984i)T + (-0.939 + 0.342i)T^{2} \)
53 \( 1 + (-1.76 - 0.642i)T + (0.766 + 0.642i)T^{2} \)
59 \( 1 + (-0.173 + 0.984i)T + (-0.939 - 0.342i)T^{2} \)
61 \( 1 + (-0.766 - 0.642i)T^{2} \)
67 \( 1 + (0.939 - 0.342i)T^{2} \)
71 \( 1 + (-0.766 + 0.642i)T^{2} \)
73 \( 1 + (-0.173 - 0.984i)T^{2} \)
79 \( 1 + (-0.173 - 0.984i)T^{2} \)
83 \( 1 + (0.5 - 0.866i)T^{2} \)
89 \( 1 + (-1.17 + 0.984i)T + (0.173 - 0.984i)T^{2} \)
97 \( 1 + (0.939 + 0.342i)T^{2} \)
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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

−9.021901059314214538043786701329, −7.891457338490927725025834123085, −7.49104343354393624675865536338, −6.94549506706901215579479197078, −6.16193680918752231056092786398, −4.87478005986949804909436223498, −4.19783122384452092203941363153, −3.55363924755681795579210301510, −2.61321291617011192515873298522, −1.03919453577127541743804489423, 0.64067467067672329575512272227, 2.52168010905966672850359869151, 3.06443301939278088646734412685, 4.07756069142930175603478358339, 5.01824292005337655078145763847, 5.71676059993186316216817789727, 6.70639589959394736297046479084, 7.14276302368859511914308675213, 8.246175633957430755118627681121, 8.976997806834584138355982580804

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