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Properties

Label 2-728-728.571-c0-0-1
Degree 22
Conductor 728728
Sign 0.400−0.916i0.400 - 0.916i
Analytic cond. 0.3633190.363319
Root an. cond. 0.6027590.602759
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 + (−0.766 + 1.32i)3-s + (−0.499 + 0.866i)4-s + (0.939 + 1.62i)5-s + 1.53·6-s + (0.766 − 0.642i)7-s + 0.999·8-s + (−0.673 − 1.16i)9-s + (0.939 − 1.62i)10-s + (−0.766 − 1.32i)12-s + 13-s + (−0.939 − 0.342i)14-s − 2.87·15-s + (−0.5 − 0.866i)16-s + (−0.173 + 0.300i)17-s + (−0.673 + 1.16i)18-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)2-s + (−0.766 + 1.32i)3-s + (−0.499 + 0.866i)4-s + (0.939 + 1.62i)5-s + 1.53·6-s + (0.766 − 0.642i)7-s + 0.999·8-s + (−0.673 − 1.16i)9-s + (0.939 − 1.62i)10-s + (−0.766 − 1.32i)12-s + 13-s + (−0.939 − 0.342i)14-s − 2.87·15-s + (−0.5 − 0.866i)16-s + (−0.173 + 0.300i)17-s + (−0.673 + 1.16i)18-s + ⋯

Functional equation

Λ(s)=(728s/2 ΓC(s) L(s)=((0.400−0.916i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.400 - 0.916i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(728s/2 ΓC(s) L(s)=((0.400−0.916i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.400 - 0.916i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 728728    =    23⋅7⋅132^{3} \cdot 7 \cdot 13
Sign: 0.400−0.916i0.400 - 0.916i
Analytic conductor: 0.3633190.363319
Root analytic conductor: 0.6027590.602759
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ728(571,⋅)\chi_{728} (571, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 728, ( :0), 0.400−0.916i)(2,\ 728,\ (\ :0),\ 0.400 - 0.916i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.71459417700.7145941770
L(12)L(\frac12) ≈\approx 0.71459417700.7145941770
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
7 1+(−0.766+0.642i)T 1 + (-0.766 + 0.642i)T
13 1−T 1 - T
good3 1+(0.766−1.32i)T+(−0.5−0.866i)T2 1 + (0.766 - 1.32i)T + (-0.5 - 0.866i)T^{2}
5 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+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
17 1+(0.173−0.300i)T+(−0.5−0.866i)T2 1 + (0.173 - 0.300i)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−T2 1 - T^{2}
31 1+(−0.5+0.866i)T+(−0.5−0.866i)T2 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2}
37 1+(0.766+1.32i)T+(−0.5+0.866i)T2 1 + (0.766 + 1.32i)T + (-0.5 + 0.866i)T^{2}
41 1−T2 1 - T^{2}
43 1+1.87T+T2 1 + 1.87T + T^{2}
47 1+(0.173+0.300i)T+(−0.5+0.866i)T2 1 + (0.173 + 0.300i)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 1−0.347T+T2 1 - 0.347T + 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−T2 1 - T^{2}
89 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
97 1−T2 1 - 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

−10.76652306429724869938642551776, −10.15876021358364465108240169645, −9.592957165843643450678748654411, −8.480944902670177016379999090219, −7.28571964248952547031922588487, −6.27055106705662348541948202154, −5.23598991571887818740669365644, −4.06218805554637986092414346063, −3.34649718180602442471678241961, −1.92580869753063113066288150171, 1.18013923071166045748001467308, 1.79876621633891232760747008070, 4.76771858041318593856184519753, 5.32102613311105753897672002550, 6.07639122919980987448816085856, 6.75489963167013748081315921839, 8.083546361754245083883054230087, 8.467815631710911945227530890184, 9.229550266598375253458869559798, 10.29404243469278849743725374537

Graph of the ZZ-function along the critical line