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Properties

Label 2-504-1.1-c5-0-31
Degree 22
Conductor 504504
Sign −1-1
Analytic cond. 80.833480.8334
Root an. cond. 8.990748.99074
Motivic weight 55
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 23.7·5-s + 49·7-s − 590.·11-s + 505.·13-s − 517.·17-s + 932.·19-s + 3.55e3·23-s − 2.55e3·25-s − 5.39e3·29-s − 8.32e3·31-s + 1.16e3·35-s − 4.93e3·37-s − 5.88e3·41-s + 1.35e4·43-s + 7.35e3·47-s + 2.40e3·49-s + 3.34e3·53-s − 1.40e4·55-s − 4.26e4·59-s + 5.13e4·61-s + 1.20e4·65-s − 3.37e4·67-s + 1.78e4·71-s − 2.06e4·73-s − 2.89e4·77-s + 9.92e4·79-s − 6.12e4·83-s + ⋯
L(s)  = 1  + 0.425·5-s + 0.377·7-s − 1.47·11-s + 0.828·13-s − 0.434·17-s + 0.592·19-s + 1.39·23-s − 0.819·25-s − 1.19·29-s − 1.55·31-s + 0.160·35-s − 0.592·37-s − 0.546·41-s + 1.11·43-s + 0.485·47-s + 0.142·49-s + 0.163·53-s − 0.626·55-s − 1.59·59-s + 1.76·61-s + 0.352·65-s − 0.918·67-s + 0.420·71-s − 0.453·73-s − 0.556·77-s + 1.78·79-s − 0.975·83-s + ⋯

Functional equation

Λ(s)=(504s/2 ΓC(s) L(s)=(− Λ(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}
Λ(s)=(504s/2 ΓC(s+5/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 504504    =    23⋅32⋅72^{3} \cdot 3^{2} \cdot 7
Sign: −1-1
Analytic conductor: 80.833480.8334
Root analytic conductor: 8.990748.99074
Motivic weight: 55
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 504, ( :5/2), −1)(2,\ 504,\ (\ :5/2),\ -1)

Particular Values

L(3)L(3) == 00
L(12)L(\frac12) == 00
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 1
7 1−49T 1 - 49T
good5 1−23.7T+3.12e3T2 1 - 23.7T + 3.12e3T^{2}
11 1+590.T+1.61e5T2 1 + 590.T + 1.61e5T^{2}
13 1−505.T+3.71e5T2 1 - 505.T + 3.71e5T^{2}
17 1+517.T+1.41e6T2 1 + 517.T + 1.41e6T^{2}
19 1−932.T+2.47e6T2 1 - 932.T + 2.47e6T^{2}
23 1−3.55e3T+6.43e6T2 1 - 3.55e3T + 6.43e6T^{2}
29 1+5.39e3T+2.05e7T2 1 + 5.39e3T + 2.05e7T^{2}
31 1+8.32e3T+2.86e7T2 1 + 8.32e3T + 2.86e7T^{2}
37 1+4.93e3T+6.93e7T2 1 + 4.93e3T + 6.93e7T^{2}
41 1+5.88e3T+1.15e8T2 1 + 5.88e3T + 1.15e8T^{2}
43 1−1.35e4T+1.47e8T2 1 - 1.35e4T + 1.47e8T^{2}
47 1−7.35e3T+2.29e8T2 1 - 7.35e3T + 2.29e8T^{2}
53 1−3.34e3T+4.18e8T2 1 - 3.34e3T + 4.18e8T^{2}
59 1+4.26e4T+7.14e8T2 1 + 4.26e4T + 7.14e8T^{2}
61 1−5.13e4T+8.44e8T2 1 - 5.13e4T + 8.44e8T^{2}
67 1+3.37e4T+1.35e9T2 1 + 3.37e4T + 1.35e9T^{2}
71 1−1.78e4T+1.80e9T2 1 - 1.78e4T + 1.80e9T^{2}
73 1+2.06e4T+2.07e9T2 1 + 2.06e4T + 2.07e9T^{2}
79 1−9.92e4T+3.07e9T2 1 - 9.92e4T + 3.07e9T^{2}
83 1+6.12e4T+3.93e9T2 1 + 6.12e4T + 3.93e9T^{2}
89 1+9.01e4T+5.58e9T2 1 + 9.01e4T + 5.58e9T^{2}
97 1+1.55e5T+8.58e9T2 1 + 1.55e5T + 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

−9.630342746918768386974113426674, −8.839494181144205098632858422830, −7.83796612066529403301010843215, −7.05776559915921493718236116441, −5.70371375546467631376438504720, −5.18763947082605368883341084022, −3.80922923162907778041928421284, −2.60476349088923990572193470942, −1.48168821974689201262425074880, 0, 1.48168821974689201262425074880, 2.60476349088923990572193470942, 3.80922923162907778041928421284, 5.18763947082605368883341084022, 5.70371375546467631376438504720, 7.05776559915921493718236116441, 7.83796612066529403301010843215, 8.839494181144205098632858422830, 9.630342746918768386974113426674

Graph of the ZZ-function along the critical line