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Properties

Label 2-588-147.23-c0-0-0
Degree 22
Conductor 588588
Sign 0.871+0.490i0.871 + 0.490i
Analytic cond. 0.2934500.293450
Root an. cond. 0.5417100.541710
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  + (0.955 − 0.294i)3-s + (0.0747 − 0.997i)7-s + (0.826 − 0.563i)9-s + (−1.72 + 0.829i)13-s + (−0.0747 + 0.129i)19-s + (−0.222 − 0.974i)21-s + (0.0747 + 0.997i)25-s + (0.623 − 0.781i)27-s + (0.988 + 1.71i)31-s + (−0.722 + 0.108i)37-s + (−1.40 + 1.29i)39-s + (−0.367 − 1.61i)43-s + (−0.988 − 0.149i)49-s + (−0.0332 + 0.145i)57-s + (−1.23 + 0.185i)61-s + ⋯
L(s)  = 1  + (0.955 − 0.294i)3-s + (0.0747 − 0.997i)7-s + (0.826 − 0.563i)9-s + (−1.72 + 0.829i)13-s + (−0.0747 + 0.129i)19-s + (−0.222 − 0.974i)21-s + (0.0747 + 0.997i)25-s + (0.623 − 0.781i)27-s + (0.988 + 1.71i)31-s + (−0.722 + 0.108i)37-s + (−1.40 + 1.29i)39-s + (−0.367 − 1.61i)43-s + (−0.988 − 0.149i)49-s + (−0.0332 + 0.145i)57-s + (−1.23 + 0.185i)61-s + ⋯

Functional equation

Λ(s)=(588s/2 ΓC(s) L(s)=((0.871+0.490i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.871 + 0.490i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(588s/2 ΓC(s) L(s)=((0.871+0.490i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.871 + 0.490i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 588588    =    22⋅3⋅722^{2} \cdot 3 \cdot 7^{2}
Sign: 0.871+0.490i0.871 + 0.490i
Analytic conductor: 0.2934500.293450
Root analytic conductor: 0.5417100.541710
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ588(317,⋅)\chi_{588} (317, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 588, ( :0), 0.871+0.490i)(2,\ 588,\ (\ :0),\ 0.871 + 0.490i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.1808953741.180895374
L(12)L(\frac12) ≈\approx 1.1808953741.180895374
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
3 1+(−0.955+0.294i)T 1 + (-0.955 + 0.294i)T
7 1+(−0.0747+0.997i)T 1 + (-0.0747 + 0.997i)T
good5 1+(−0.0747−0.997i)T2 1 + (-0.0747 - 0.997i)T^{2}
11 1+(−0.365−0.930i)T2 1 + (-0.365 - 0.930i)T^{2}
13 1+(1.72−0.829i)T+(0.623−0.781i)T2 1 + (1.72 - 0.829i)T + (0.623 - 0.781i)T^{2}
17 1+(0.733+0.680i)T2 1 + (0.733 + 0.680i)T^{2}
19 1+(0.0747−0.129i)T+(−0.5−0.866i)T2 1 + (0.0747 - 0.129i)T + (-0.5 - 0.866i)T^{2}
23 1+(0.733−0.680i)T2 1 + (0.733 - 0.680i)T^{2}
29 1+(0.222−0.974i)T2 1 + (0.222 - 0.974i)T^{2}
31 1+(−0.988−1.71i)T+(−0.5+0.866i)T2 1 + (-0.988 - 1.71i)T + (-0.5 + 0.866i)T^{2}
37 1+(0.722−0.108i)T+(0.955−0.294i)T2 1 + (0.722 - 0.108i)T + (0.955 - 0.294i)T^{2}
41 1+(0.900+0.433i)T2 1 + (0.900 + 0.433i)T^{2}
43 1+(0.367+1.61i)T+(−0.900+0.433i)T2 1 + (0.367 + 1.61i)T + (-0.900 + 0.433i)T^{2}
47 1+(0.988+0.149i)T2 1 + (0.988 + 0.149i)T^{2}
53 1+(−0.955−0.294i)T2 1 + (-0.955 - 0.294i)T^{2}
59 1+(−0.0747+0.997i)T2 1 + (-0.0747 + 0.997i)T^{2}
61 1+(1.23−0.185i)T+(0.955−0.294i)T2 1 + (1.23 - 0.185i)T + (0.955 - 0.294i)T^{2}
67 1+(−0.733−1.26i)T+(−0.5+0.866i)T2 1 + (-0.733 - 1.26i)T + (-0.5 + 0.866i)T^{2}
71 1+(0.222+0.974i)T2 1 + (0.222 + 0.974i)T^{2}
73 1+(−0.0546−0.728i)T+(−0.988+0.149i)T2 1 + (-0.0546 - 0.728i)T + (-0.988 + 0.149i)T^{2}
79 1+(−0.988+1.71i)T+(−0.5−0.866i)T2 1 + (-0.988 + 1.71i)T + (-0.5 - 0.866i)T^{2}
83 1+(−0.623−0.781i)T2 1 + (-0.623 - 0.781i)T^{2}
89 1+(−0.365+0.930i)T2 1 + (-0.365 + 0.930i)T^{2}
97 1+1.80T+T2 1 + 1.80T + 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

−10.59266794104363523535901504593, −9.901336312715754622516277214113, −9.101983417021709464379179872515, −8.149942581575917872035370934960, −7.13093244527214949503733897023, −6.87950073630855021649770342849, −5.06381654486377712037030853294, −4.12137799058218943481813301017, −2.99851932164108026702505154775, −1.67903677522444305254266723655, 2.26525933591875023635767300839, 2.95331654866925326455233834708, 4.43978048812425102550902004829, 5.26957849517628938219726786245, 6.53160259441540444603463224388, 7.80213804058705718923847174302, 8.208362343833780376133810583147, 9.417986382029737363457824390255, 9.788237941107163711228820923545, 10.81108296844765077789176212965

Graph of the ZZ-function along the critical line