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Properties

Label 2-115-1.1-c5-0-15
Degree 22
Conductor 115115
Sign 11
Analytic cond. 18.444118.4441
Root an. cond. 4.294664.29466
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  − 8.44·2-s + 28.1·3-s + 39.2·4-s + 25·5-s − 237.·6-s − 83.6·7-s − 61.3·8-s + 550.·9-s − 211.·10-s + 87.3·11-s + 1.10e3·12-s + 920.·13-s + 705.·14-s + 704.·15-s − 738.·16-s − 712.·17-s − 4.65e3·18-s + 728.·19-s + 981.·20-s − 2.35e3·21-s − 737.·22-s − 529·23-s − 1.72e3·24-s + 625·25-s − 7.77e3·26-s + 8.67e3·27-s − 3.28e3·28-s + ⋯
L(s)  = 1  − 1.49·2-s + 1.80·3-s + 1.22·4-s + 0.447·5-s − 2.69·6-s − 0.645·7-s − 0.338·8-s + 2.26·9-s − 0.667·10-s + 0.217·11-s + 2.21·12-s + 1.51·13-s + 0.962·14-s + 0.808·15-s − 0.721·16-s − 0.598·17-s − 3.38·18-s + 0.463·19-s + 0.548·20-s − 1.16·21-s − 0.324·22-s − 0.208·23-s − 0.612·24-s + 0.200·25-s − 2.25·26-s + 2.28·27-s − 0.791·28-s + ⋯

Functional equation

Λ(s)=(115s/2 ΓC(s) L(s)=( Λ(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}
Λ(s)=(115s/2 ΓC(s+5/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 115115    =    5⋅235 \cdot 23
Sign: 11
Analytic conductor: 18.444118.4441
Root analytic conductor: 4.294664.29466
Motivic weight: 55
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 115, ( :5/2), 1)(2,\ 115,\ (\ :5/2),\ 1)

Particular Values

L(3)L(3) ≈\approx 1.8724862001.872486200
L(12)L(\frac12) ≈\approx 1.8724862001.872486200
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)
bad5 1−25T 1 - 25T
23 1+529T 1 + 529T
good2 1+8.44T+32T2 1 + 8.44T + 32T^{2}
3 1−28.1T+243T2 1 - 28.1T + 243T^{2}
7 1+83.6T+1.68e4T2 1 + 83.6T + 1.68e4T^{2}
11 1−87.3T+1.61e5T2 1 - 87.3T + 1.61e5T^{2}
13 1−920.T+3.71e5T2 1 - 920.T + 3.71e5T^{2}
17 1+712.T+1.41e6T2 1 + 712.T + 1.41e6T^{2}
19 1−728.T+2.47e6T2 1 - 728.T + 2.47e6T^{2}
29 1−1.71e3T+2.05e7T2 1 - 1.71e3T + 2.05e7T^{2}
31 1+5.78e3T+2.86e7T2 1 + 5.78e3T + 2.86e7T^{2}
37 1−1.01e4T+6.93e7T2 1 - 1.01e4T + 6.93e7T^{2}
41 1−1.80e4T+1.15e8T2 1 - 1.80e4T + 1.15e8T^{2}
43 1−536.T+1.47e8T2 1 - 536.T + 1.47e8T^{2}
47 1+1.43e4T+2.29e8T2 1 + 1.43e4T + 2.29e8T^{2}
53 1−2.54e4T+4.18e8T2 1 - 2.54e4T + 4.18e8T^{2}
59 1−2.33e4T+7.14e8T2 1 - 2.33e4T + 7.14e8T^{2}
61 1−2.39e4T+8.44e8T2 1 - 2.39e4T + 8.44e8T^{2}
67 1+6.41e4T+1.35e9T2 1 + 6.41e4T + 1.35e9T^{2}
71 1−5.76e4T+1.80e9T2 1 - 5.76e4T + 1.80e9T^{2}
73 1−2.60e4T+2.07e9T2 1 - 2.60e4T + 2.07e9T^{2}
79 1−6.08e3T+3.07e9T2 1 - 6.08e3T + 3.07e9T^{2}
83 1+5.37e4T+3.93e9T2 1 + 5.37e4T + 3.93e9T^{2}
89 1−1.09e5T+5.58e9T2 1 - 1.09e5T + 5.58e9T^{2}
97 1+1.45e5T+8.58e9T2 1 + 1.45e5T + 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

−13.01625713524376543774342482654, −11.06955140584552353143090490522, −9.933635539456683194154542764298, −9.229292674333859529923699440501, −8.608302820811833076677491374338, −7.65009556931469228593669870368, −6.49489885265259908547923857644, −3.86312064518784024257907871651, −2.47420444457956940407434633430, −1.21980223088510120628662331946, 1.21980223088510120628662331946, 2.47420444457956940407434633430, 3.86312064518784024257907871651, 6.49489885265259908547923857644, 7.65009556931469228593669870368, 8.608302820811833076677491374338, 9.229292674333859529923699440501, 9.933635539456683194154542764298, 11.06955140584552353143090490522, 13.01625713524376543774342482654

Graph of the ZZ-function along the critical line