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Properties

Label 2-1260-1260.299-c0-0-3
Degree 22
Conductor 12601260
Sign −0.296+0.954i-0.296 + 0.954i
Analytic cond. 0.6288210.628821
Root an. cond. 0.7929820.792982
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  + (0.866 − 0.5i)2-s + (0.866 − 0.5i)3-s + (0.499 − 0.866i)4-s − 5-s + (0.499 − 0.866i)6-s + (−0.866 − 0.5i)7-s − 0.999i·8-s + (0.499 − 0.866i)9-s + (−0.866 + 0.5i)10-s − 0.999i·12-s − 0.999·14-s + (−0.866 + 0.5i)15-s + (−0.5 − 0.866i)16-s − 0.999i·18-s + (−0.499 + 0.866i)20-s − 0.999·21-s + ⋯
L(s)  = 1  + (0.866 − 0.5i)2-s + (0.866 − 0.5i)3-s + (0.499 − 0.866i)4-s − 5-s + (0.499 − 0.866i)6-s + (−0.866 − 0.5i)7-s − 0.999i·8-s + (0.499 − 0.866i)9-s + (−0.866 + 0.5i)10-s − 0.999i·12-s − 0.999·14-s + (−0.866 + 0.5i)15-s + (−0.5 − 0.866i)16-s − 0.999i·18-s + (−0.499 + 0.866i)20-s − 0.999·21-s + ⋯

Functional equation

Λ(s)=(1260s/2 ΓC(s) L(s)=((−0.296+0.954i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.296 + 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(1260s/2 ΓC(s) L(s)=((−0.296+0.954i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.296 + 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 12601260    =    22⋅32⋅5⋅72^{2} \cdot 3^{2} \cdot 5 \cdot 7
Sign: −0.296+0.954i-0.296 + 0.954i
Analytic conductor: 0.6288210.628821
Root analytic conductor: 0.7929820.792982
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ1260(299,⋅)\chi_{1260} (299, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1260, ( :0), −0.296+0.954i)(2,\ 1260,\ (\ :0),\ -0.296 + 0.954i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.8043780351.804378035
L(12)L(\frac12) ≈\approx 1.8043780351.804378035
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+(−0.866+0.5i)T 1 + (-0.866 + 0.5i)T
3 1+(−0.866+0.5i)T 1 + (-0.866 + 0.5i)T
5 1+T 1 + T
7 1+(0.866+0.5i)T 1 + (0.866 + 0.5i)T
good11 1+T2 1 + T^{2}
13 1+(−0.5+0.866i)T2 1 + (-0.5 + 0.866i)T^{2}
17 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
19 1+(−0.5−0.866i)T2 1 + (-0.5 - 0.866i)T^{2}
23 1−2iT−T2 1 - 2iT - T^{2}
29 1+(−1.5−0.866i)T+(0.5+0.866i)T2 1 + (-1.5 - 0.866i)T + (0.5 + 0.866i)T^{2}
31 1+(−0.5−0.866i)T2 1 + (-0.5 - 0.866i)T^{2}
37 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
41 1+(−0.5−0.866i)T+(−0.5+0.866i)T2 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2}
43 1+(−0.866+1.5i)T+(−0.5−0.866i)T2 1 + (-0.866 + 1.5i)T + (-0.5 - 0.866i)T^{2}
47 1+(0.866+1.5i)T+(−0.5+0.866i)T2 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2}
53 1+(−0.5+0.866i)T2 1 + (-0.5 + 0.866i)T^{2}
59 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
61 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
67 1+(−0.5−0.866i)T2 1 + (-0.5 - 0.866i)T^{2}
71 1+T2 1 + T^{2}
73 1+(−0.5+0.866i)T2 1 + (-0.5 + 0.866i)T^{2}
79 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
83 1+(0.866−1.5i)T+(−0.5−0.866i)T2 1 + (0.866 - 1.5i)T + (-0.5 - 0.866i)T^{2}
89 1+(1−1.73i)T+(−0.5−0.866i)T2 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2}
97 1+(−0.5−0.866i)T2 1 + (-0.5 - 0.866i)T^{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

−9.722903308503074943187783227164, −8.906306890734304115196595285779, −7.83402656602402720356312652080, −7.08306263283818787325872646738, −6.52896801410802086717922928816, −5.25932466250393343784622561448, −4.03295640683285972302453678291, −3.52041502107327942795861360787, −2.70895887295229795491606934733, −1.18522564287751680934537943335, 2.60121535867422162821335173420, 3.10502512536619287865005350163, 4.24942676342106737749656615011, 4.65159310679076219922108652710, 6.02831129149565488126926598592, 6.80200915222145220340710965467, 7.74017106697005777442533150196, 8.396080466477199243907773762826, 9.021106584700497471244897484133, 10.12138138219703362379779919558

Graph of the ZZ-function along the critical line