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Properties

Label 2-588-1.1-c5-0-7
Degree 22
Conductor 588588
Sign 11
Analytic cond. 94.305694.3056
Root an. cond. 9.711119.71111
Motivic weight 55
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 9·3-s − 78.6·5-s + 81·9-s + 691.·11-s − 818.·13-s − 708.·15-s − 1.10e3·17-s + 573.·19-s + 2.51e3·23-s + 3.06e3·25-s + 729·27-s − 3.25e3·29-s − 1.01e4·31-s + 6.22e3·33-s + 4.86e3·37-s − 7.36e3·39-s + 1.30e4·41-s − 9.30e3·43-s − 6.37e3·45-s − 1.29e4·47-s − 9.98e3·51-s − 1.95e4·53-s − 5.44e4·55-s + 5.15e3·57-s + 2.51e4·59-s + 3.13e4·61-s + 6.44e4·65-s + ⋯
L(s)  = 1  + 0.577·3-s − 1.40·5-s + 0.333·9-s + 1.72·11-s − 1.34·13-s − 0.812·15-s − 0.930·17-s + 0.364·19-s + 0.992·23-s + 0.980·25-s + 0.192·27-s − 0.719·29-s − 1.89·31-s + 0.994·33-s + 0.584·37-s − 0.775·39-s + 1.21·41-s − 0.767·43-s − 0.469·45-s − 0.852·47-s − 0.537·51-s − 0.955·53-s − 2.42·55-s + 0.210·57-s + 0.939·59-s + 1.07·61-s + 1.89·65-s + ⋯

Functional equation

Λ(s)=(588s/2 ΓC(s) L(s)=( Λ(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}
Λ(s)=(588s/2 ΓC(s+5/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 588588    =    22⋅3⋅722^{2} \cdot 3 \cdot 7^{2}
Sign: 11
Analytic conductor: 94.305694.3056
Root analytic conductor: 9.711119.71111
Motivic weight: 55
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 588, ( :5/2), 1)(2,\ 588,\ (\ :5/2),\ 1)

Particular Values

L(3)L(3) ≈\approx 1.7665907041.766590704
L(12)L(\frac12) ≈\approx 1.7665907041.766590704
L(72)L(\frac{7}{2}) 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 1−9T 1 - 9T
7 1 1
good5 1+78.6T+3.12e3T2 1 + 78.6T + 3.12e3T^{2}
11 1−691.T+1.61e5T2 1 - 691.T + 1.61e5T^{2}
13 1+818.T+3.71e5T2 1 + 818.T + 3.71e5T^{2}
17 1+1.10e3T+1.41e6T2 1 + 1.10e3T + 1.41e6T^{2}
19 1−573.T+2.47e6T2 1 - 573.T + 2.47e6T^{2}
23 1−2.51e3T+6.43e6T2 1 - 2.51e3T + 6.43e6T^{2}
29 1+3.25e3T+2.05e7T2 1 + 3.25e3T + 2.05e7T^{2}
31 1+1.01e4T+2.86e7T2 1 + 1.01e4T + 2.86e7T^{2}
37 1−4.86e3T+6.93e7T2 1 - 4.86e3T + 6.93e7T^{2}
41 1−1.30e4T+1.15e8T2 1 - 1.30e4T + 1.15e8T^{2}
43 1+9.30e3T+1.47e8T2 1 + 9.30e3T + 1.47e8T^{2}
47 1+1.29e4T+2.29e8T2 1 + 1.29e4T + 2.29e8T^{2}
53 1+1.95e4T+4.18e8T2 1 + 1.95e4T + 4.18e8T^{2}
59 1−2.51e4T+7.14e8T2 1 - 2.51e4T + 7.14e8T^{2}
61 1−3.13e4T+8.44e8T2 1 - 3.13e4T + 8.44e8T^{2}
67 1−5.59e4T+1.35e9T2 1 - 5.59e4T + 1.35e9T^{2}
71 1−2.05e4T+1.80e9T2 1 - 2.05e4T + 1.80e9T^{2}
73 1−6.76e4T+2.07e9T2 1 - 6.76e4T + 2.07e9T^{2}
79 1−1.40e4T+3.07e9T2 1 - 1.40e4T + 3.07e9T^{2}
83 1−7.71e4T+3.93e9T2 1 - 7.71e4T + 3.93e9T^{2}
89 1+320.T+5.58e9T2 1 + 320.T + 5.58e9T^{2}
97 1−1.12e5T+8.58e9T2 1 - 1.12e5T + 8.58e9T^{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

−9.551447390934058706702438166483, −9.134877880326552358716895645280, −8.099043227170712005772906475438, −7.26860516723582170325710347288, −6.69808121762116658713841796084, −5.03250606626888387137054384718, −4.06828331719991921553191082911, −3.42814849294869845924327700258, −2.05406851904030609848654173438, −0.62230237170248182907945090259, 0.62230237170248182907945090259, 2.05406851904030609848654173438, 3.42814849294869845924327700258, 4.06828331719991921553191082911, 5.03250606626888387137054384718, 6.69808121762116658713841796084, 7.26860516723582170325710347288, 8.099043227170712005772906475438, 9.134877880326552358716895645280, 9.551447390934058706702438166483

Graph of the ZZ-function along the critical line