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Properties

Label 2-1352-104.43-c0-0-1
Degree 22
Conductor 13521352
Sign 0.711−0.702i0.711 - 0.702i
Analytic cond. 0.6747350.674735
Root an. cond. 0.8214230.821423
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 − 5-s + (0.499 + 0.866i)6-s + (0.5 + 0.866i)7-s + 0.999·8-s + (0.5 − 0.866i)10-s − 0.999·12-s − 0.999·14-s + (−0.5 + 0.866i)15-s + (−0.5 + 0.866i)16-s + (0.5 + 0.866i)17-s + (0.499 + 0.866i)20-s + 0.999·21-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)2-s + (0.5 − 0.866i)3-s + (−0.499 − 0.866i)4-s − 5-s + (0.499 + 0.866i)6-s + (0.5 + 0.866i)7-s + 0.999·8-s + (0.5 − 0.866i)10-s − 0.999·12-s − 0.999·14-s + (−0.5 + 0.866i)15-s + (−0.5 + 0.866i)16-s + (0.5 + 0.866i)17-s + (0.499 + 0.866i)20-s + 0.999·21-s + ⋯

Functional equation

Λ(s)=(1352s/2 ΓC(s) L(s)=((0.711−0.702i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.711 - 0.702i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(1352s/2 ΓC(s) L(s)=((0.711−0.702i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.711 - 0.702i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 13521352    =    23⋅1322^{3} \cdot 13^{2}
Sign: 0.711−0.702i0.711 - 0.702i
Analytic conductor: 0.6747350.674735
Root analytic conductor: 0.8214230.821423
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ1352(147,⋅)\chi_{1352} (147, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1352, ( :0), 0.711−0.702i)(2,\ 1352,\ (\ :0),\ 0.711 - 0.702i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.86895757830.8689575783
L(12)L(\frac12) ≈\approx 0.86895757830.8689575783
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
13 1 1
good3 1+(−0.5+0.866i)T+(−0.5−0.866i)T2 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2}
5 1+T+T2 1 + T + T^{2}
7 1+(−0.5−0.866i)T+(−0.5+0.866i)T2 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2}
11 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
17 1+(−0.5−0.866i)T+(−0.5+0.866i)T2 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2}
19 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
23 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
29 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
31 1−2T+T2 1 - 2T + T^{2}
37 1+(−0.5+0.866i)T+(−0.5−0.866i)T2 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2}
41 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
43 1+(−0.5−0.866i)T+(−0.5+0.866i)T2 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2}
47 1+T+T2 1 + T + T^{2}
53 1−T2 1 - 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+(−0.5−0.866i)T+(−0.5+0.866i)T2 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2}
73 1−T2 1 - T^{2}
79 1−T2 1 - T^{2}
83 1−T2 1 - T^{2}
89 1+(0.5+0.866i)T2 1 + (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

−9.639883151340193740955985166158, −8.637990824210638062983306887919, −8.061293133847610931600101169269, −7.81243000993520649585949784565, −6.82256053180735567459924108778, −6.01705003682056279250632511020, −4.99992003657759727172325455197, −4.05037507740436950626742637493, −2.54358603099472396009383018560, −1.33459080947564551581067895464, 1.00953677988744670395117215108, 2.77379702019684074899418429618, 3.60941195263442396701033604260, 4.30830815097083046854960840124, 4.89295158604076564450827586576, 6.74771899739252605207314424248, 7.71871737014065492320530361274, 8.142546902488715057187245317363, 9.037523255936515794198935074887, 9.855339659553033433900507536411

Graph of the ZZ-function along the critical line