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Properties

Label 2-2960-2960.1173-c0-0-0
Degree 22
Conductor 29602960
Sign 0.888−0.459i0.888 - 0.459i
Analytic cond. 1.477231.47723
Root an. cond. 1.215411.21541
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 + 0.866i)2-s + (0.5 − 0.866i)3-s + (−0.499 + 0.866i)4-s + (0.5 − 0.866i)5-s + 0.999·6-s + (1.36 − 0.366i)7-s − 0.999·8-s + 0.999·10-s + (1 + i)11-s + (0.499 + 0.866i)12-s + (−0.5 + 0.866i)13-s + (1 + 0.999i)14-s + (−0.499 − 0.866i)15-s + (−0.5 − 0.866i)16-s + (0.499 + 0.866i)20-s + (0.366 − 1.36i)21-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)2-s + (0.5 − 0.866i)3-s + (−0.499 + 0.866i)4-s + (0.5 − 0.866i)5-s + 0.999·6-s + (1.36 − 0.366i)7-s − 0.999·8-s + 0.999·10-s + (1 + i)11-s + (0.499 + 0.866i)12-s + (−0.5 + 0.866i)13-s + (1 + 0.999i)14-s + (−0.499 − 0.866i)15-s + (−0.5 − 0.866i)16-s + (0.499 + 0.866i)20-s + (0.366 − 1.36i)21-s + ⋯

Functional equation

Λ(s)=(2960s/2 ΓC(s) L(s)=((0.888−0.459i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.888 - 0.459i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(2960s/2 ΓC(s) L(s)=((0.888−0.459i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.888 - 0.459i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 29602960    =    24⋅5⋅372^{4} \cdot 5 \cdot 37
Sign: 0.888−0.459i0.888 - 0.459i
Analytic conductor: 1.477231.47723
Root analytic conductor: 1.215411.21541
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ2960(1173,⋅)\chi_{2960} (1173, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2960, ( :0), 0.888−0.459i)(2,\ 2960,\ (\ :0),\ 0.888 - 0.459i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 2.2667657042.266765704
L(12)L(\frac12) ≈\approx 2.2667657042.266765704
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+(−0.5−0.866i)T 1 + (-0.5 - 0.866i)T
5 1+(−0.5+0.866i)T 1 + (-0.5 + 0.866i)T
37 1+(0.5−0.866i)T 1 + (0.5 - 0.866i)T
good3 1+(−0.5+0.866i)T+(−0.5−0.866i)T2 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2}
7 1+(−1.36+0.366i)T+(0.866−0.5i)T2 1 + (-1.36 + 0.366i)T + (0.866 - 0.5i)T^{2}
11 1+(−1−i)T+iT2 1 + (-1 - i)T + iT^{2}
13 1+(0.5−0.866i)T+(−0.5−0.866i)T2 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2}
17 1+(0.866+0.5i)T2 1 + (0.866 + 0.5i)T^{2}
19 1+(0.866−0.5i)T2 1 + (0.866 - 0.5i)T^{2}
23 1+(1−i)T−iT2 1 + (1 - i)T - iT^{2}
29 1+iT2 1 + iT^{2}
31 1+T+T2 1 + T + T^{2}
41 1+(0.866+0.5i)T+(0.5+0.866i)T2 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2}
43 1+iT−T2 1 + iT - T^{2}
47 1−iT2 1 - iT^{2}
53 1+(−0.866+0.5i)T+(0.5−0.866i)T2 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2}
59 1+(1.36+0.366i)T+(0.866+0.5i)T2 1 + (1.36 + 0.366i)T + (0.866 + 0.5i)T^{2}
61 1+(0.366+1.36i)T+(−0.866+0.5i)T2 1 + (0.366 + 1.36i)T + (-0.866 + 0.5i)T^{2}
67 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
71 1+(1.73+i)T+(0.5+0.866i)T2 1 + (1.73 + i)T + (0.5 + 0.866i)T^{2}
73 1−iT2 1 - iT^{2}
79 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
83 1+(−0.5+0.866i)T2 1 + (-0.5 + 0.866i)T^{2}
89 1+(−0.5+0.866i)T2 1 + (-0.5 + 0.866i)T^{2}
97 1−iT2 1 - iT^{2}
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   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

−8.744918097350213145352742707939, −8.061436236222512281588085301376, −7.39543012121670606473748436837, −6.96433300783165662911454341348, −5.99764561758293062944112577559, −4.94522656567518875200642168861, −4.62262723815246689999144778096, −3.70871764324636829082618824467, −1.92000757654363099373870494719, −1.70567787575489636158966377567, 1.43355795155110563840772398439, 2.48009492137445722880490452763, 3.24186271827288508707532292692, 4.01062486430068073345701601785, 4.77196016251556714169107841172, 5.68385665024294844466609732940, 6.23657639029656423688377238112, 7.43340238240120863292042633022, 8.548031844796485534670684792423, 8.929135413885952907577493325162

Graph of the ZZ-function along the critical line