↑

Properties

Label 2-3040-760.579-c0-0-0
Degree 22
Conductor 30403040
Sign 0.776+0.630i0.776 + 0.630i
Analytic cond. 1.517151.51715
Root an. cond. 1.231721.23172
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

Related objects

Downloads

Learn more

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

Λ(s)=(3040s/2 ΓC(s) L(s)=((0.776+0.630i) Λ‾(1−s)\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}
Λ(s)=(3040s/2 ΓC(s) L(s)=((0.776+0.630i) Λ‾(1−s)\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: 22
Conductor: 30403040    =    25⋅5⋅192^{5} \cdot 5 \cdot 19
Sign: 0.776+0.630i0.776 + 0.630i
Analytic conductor: 1.517151.51715
Root analytic conductor: 1.231721.23172
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ3040(2479,⋅)\chi_{3040} (2479, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3040, ( :0), 0.776+0.630i)(2,\ 3040,\ (\ :0),\ 0.776 + 0.630i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.82836322520.8283632252
L(12)L(\frac12) ≈\approx 0.82836322520.8283632252
L(1)L(1) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1 1
5 1+(0.939+0.342i)T 1 + (0.939 + 0.342i)T
19 1+(0.173+0.984i)T 1 + (0.173 + 0.984i)T
good3 1+(−0.173+0.984i)T2 1 + (-0.173 + 0.984i)T^{2}
7 1+(0.939−1.62i)T+(−0.5−0.866i)T2 1 + (0.939 - 1.62i)T + (-0.5 - 0.866i)T^{2}
11 1+(−0.173−0.300i)T+(−0.5+0.866i)T2 1 + (-0.173 - 0.300i)T + (-0.5 + 0.866i)T^{2}
13 1+(0.766+0.642i)T+(0.173+0.984i)T2 1 + (0.766 + 0.642i)T + (0.173 + 0.984i)T^{2}
17 1+(0.939−0.342i)T2 1 + (0.939 - 0.342i)T^{2}
23 1+(−1.43+0.524i)T+(0.766−0.642i)T2 1 + (-1.43 + 0.524i)T + (0.766 - 0.642i)T^{2}
29 1+(0.939+0.342i)T2 1 + (0.939 + 0.342i)T^{2}
31 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
37 1−1.53T+T2 1 - 1.53T + T^{2}
41 1+(−0.266+0.223i)T+(0.173−0.984i)T2 1 + (-0.266 + 0.223i)T + (0.173 - 0.984i)T^{2}
43 1+(−0.766−0.642i)T2 1 + (-0.766 - 0.642i)T^{2}
47 1+(−0.173+0.984i)T+(−0.939−0.342i)T2 1 + (-0.173 + 0.984i)T + (-0.939 - 0.342i)T^{2}
53 1+(−1.76+0.642i)T+(0.766−0.642i)T2 1 + (-1.76 + 0.642i)T + (0.766 - 0.642i)T^{2}
59 1+(−0.173−0.984i)T+(−0.939+0.342i)T2 1 + (-0.173 - 0.984i)T + (-0.939 + 0.342i)T^{2}
61 1+(−0.766+0.642i)T2 1 + (-0.766 + 0.642i)T^{2}
67 1+(0.939+0.342i)T2 1 + (0.939 + 0.342i)T^{2}
71 1+(−0.766−0.642i)T2 1 + (-0.766 - 0.642i)T^{2}
73 1+(−0.173+0.984i)T2 1 + (-0.173 + 0.984i)T^{2}
79 1+(−0.173+0.984i)T2 1 + (-0.173 + 0.984i)T^{2}
83 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
89 1+(−1.17−0.984i)T+(0.173+0.984i)T2 1 + (-1.17 - 0.984i)T + (0.173 + 0.984i)T^{2}
97 1+(0.939−0.342i)T2 1 + (0.939 - 0.342i)T^{2}
show more
show less
   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(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

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

Graph of the ZZ-function along the critical line