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Properties

Label 2-592-1.1-c1-0-15
Degree 22
Conductor 592592
Sign −1-1
Analytic cond. 4.727144.72714
Root an. cond. 2.174192.17419
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 2·5-s − 7-s − 2·9-s − 11-s − 6·13-s − 2·15-s − 4·17-s + 8·19-s − 21-s − 6·23-s − 25-s − 5·27-s + 2·29-s + 4·31-s − 33-s + 2·35-s − 37-s − 6·39-s + 7·41-s − 2·43-s + 4·45-s − 9·47-s − 6·49-s − 4·51-s − 3·53-s + 2·55-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.894·5-s − 0.377·7-s − 2/3·9-s − 0.301·11-s − 1.66·13-s − 0.516·15-s − 0.970·17-s + 1.83·19-s − 0.218·21-s − 1.25·23-s − 1/5·25-s − 0.962·27-s + 0.371·29-s + 0.718·31-s − 0.174·33-s + 0.338·35-s − 0.164·37-s − 0.960·39-s + 1.09·41-s − 0.304·43-s + 0.596·45-s − 1.31·47-s − 6/7·49-s − 0.560·51-s − 0.412·53-s + 0.269·55-s + ⋯

Functional equation

Λ(s)=(592s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(592s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 592 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 592592    =    24⋅372^{4} \cdot 37
Sign: −1-1
Analytic conductor: 4.727144.72714
Root analytic conductor: 2.174192.17419
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 592, ( :1/2), −1)(2,\ 592,\ (\ :1/2),\ -1)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
L(32)L(\frac{3}{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
37 1+T 1 + T
good3 1−T+pT2 1 - T + p T^{2}
5 1+2T+pT2 1 + 2 T + p T^{2}
7 1+T+pT2 1 + T + p T^{2}
11 1+T+pT2 1 + T + p T^{2}
13 1+6T+pT2 1 + 6 T + p T^{2}
17 1+4T+pT2 1 + 4 T + p T^{2}
19 1−8T+pT2 1 - 8 T + p T^{2}
23 1+6T+pT2 1 + 6 T + p T^{2}
29 1−2T+pT2 1 - 2 T + p T^{2}
31 1−4T+pT2 1 - 4 T + p T^{2}
41 1−7T+pT2 1 - 7 T + p T^{2}
43 1+2T+pT2 1 + 2 T + p T^{2}
47 1+9T+pT2 1 + 9 T + p T^{2}
53 1+3T+pT2 1 + 3 T + p T^{2}
59 1−12T+pT2 1 - 12 T + p T^{2}
61 1−4T+pT2 1 - 4 T + p T^{2}
67 1+pT2 1 + p T^{2}
71 1+7T+pT2 1 + 7 T + p T^{2}
73 1−7T+pT2 1 - 7 T + p T^{2}
79 1+pT2 1 + p T^{2}
83 1+3T+pT2 1 + 3 T + p T^{2}
89 1+12T+pT2 1 + 12 T + p T^{2}
97 1+8T+pT2 1 + 8 T + p 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

−9.969562582054726543474486278240, −9.494663490575910035621177927785, −8.289160116966715982503627200518, −7.75456110220713984165022091972, −6.86682095955105512061953544939, −5.55163436815551852246094638061, −4.49161097993379934705251491217, −3.32391285921883791433244095783, −2.42130817560810783959503841896, 0, 2.42130817560810783959503841896, 3.32391285921883791433244095783, 4.49161097993379934705251491217, 5.55163436815551852246094638061, 6.86682095955105512061953544939, 7.75456110220713984165022091972, 8.289160116966715982503627200518, 9.494663490575910035621177927785, 9.969562582054726543474486278240

Graph of the ZZ-function along the critical line