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Properties

Label 2-234-1.1-c5-0-18
Degree 22
Conductor 234234
Sign −1-1
Analytic cond. 37.529837.5298
Root an. cond. 6.126156.12615
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  − 4·2-s + 16·4-s + 51.9·5-s + 125.·7-s − 64·8-s − 207.·10-s − 727.·11-s + 169·13-s − 503.·14-s + 256·16-s − 1.75e3·17-s − 1.91e3·19-s + 830.·20-s + 2.90e3·22-s + 3.54e3·23-s − 428.·25-s − 676·26-s + 2.01e3·28-s − 4.86e3·29-s + 2.31e3·31-s − 1.02e3·32-s + 7.01e3·34-s + 6.53e3·35-s + 855.·37-s + 7.64e3·38-s − 3.32e3·40-s + 1.59e4·41-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.5·4-s + 0.928·5-s + 0.971·7-s − 0.353·8-s − 0.656·10-s − 1.81·11-s + 0.277·13-s − 0.686·14-s + 0.250·16-s − 1.47·17-s − 1.21·19-s + 0.464·20-s + 1.28·22-s + 1.39·23-s − 0.137·25-s − 0.196·26-s + 0.485·28-s − 1.07·29-s + 0.432·31-s − 0.176·32-s + 1.04·34-s + 0.902·35-s + 0.102·37-s + 0.859·38-s − 0.328·40-s + 1.48·41-s + ⋯

Functional equation

Λ(s)=(234s/2 ΓC(s) L(s)=(− Λ(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}
Λ(s)=(234s/2 ΓC(s+5/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 234234    =    2⋅32⋅132 \cdot 3^{2} \cdot 13
Sign: −1-1
Analytic conductor: 37.529837.5298
Root analytic conductor: 6.126156.12615
Motivic weight: 55
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 234, ( :5/2), −1)(2,\ 234,\ (\ :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+4T 1 + 4T
3 1 1
13 1−169T 1 - 169T
good5 1−51.9T+3.12e3T2 1 - 51.9T + 3.12e3T^{2}
7 1−125.T+1.68e4T2 1 - 125.T + 1.68e4T^{2}
11 1+727.T+1.61e5T2 1 + 727.T + 1.61e5T^{2}
17 1+1.75e3T+1.41e6T2 1 + 1.75e3T + 1.41e6T^{2}
19 1+1.91e3T+2.47e6T2 1 + 1.91e3T + 2.47e6T^{2}
23 1−3.54e3T+6.43e6T2 1 - 3.54e3T + 6.43e6T^{2}
29 1+4.86e3T+2.05e7T2 1 + 4.86e3T + 2.05e7T^{2}
31 1−2.31e3T+2.86e7T2 1 - 2.31e3T + 2.86e7T^{2}
37 1−855.T+6.93e7T2 1 - 855.T + 6.93e7T^{2}
41 1−1.59e4T+1.15e8T2 1 - 1.59e4T + 1.15e8T^{2}
43 1+1.82e4T+1.47e8T2 1 + 1.82e4T + 1.47e8T^{2}
47 1+2.54e4T+2.29e8T2 1 + 2.54e4T + 2.29e8T^{2}
53 1−6.92e3T+4.18e8T2 1 - 6.92e3T + 4.18e8T^{2}
59 1−2.82e4T+7.14e8T2 1 - 2.82e4T + 7.14e8T^{2}
61 1−1.70e4T+8.44e8T2 1 - 1.70e4T + 8.44e8T^{2}
67 1+6.13e4T+1.35e9T2 1 + 6.13e4T + 1.35e9T^{2}
71 1−3.47e4T+1.80e9T2 1 - 3.47e4T + 1.80e9T^{2}
73 1+5.14e4T+2.07e9T2 1 + 5.14e4T + 2.07e9T^{2}
79 1−1.02e3T+3.07e9T2 1 - 1.02e3T + 3.07e9T^{2}
83 1+1.17e5T+3.93e9T2 1 + 1.17e5T + 3.93e9T^{2}
89 1+2.44e4T+5.58e9T2 1 + 2.44e4T + 5.58e9T^{2}
97 1−3.23e4T+8.58e9T2 1 - 3.23e4T + 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

−10.83210992893769142493137773987, −9.906001270915681546609300347089, −8.775689139800608709658075136895, −8.065877480422518827358619720284, −6.89444137509146616852745549600, −5.69510265828470869796838823837, −4.68417151196206501694114695562, −2.60120610251849185115840731512, −1.73471197844421914850196786930, 0, 1.73471197844421914850196786930, 2.60120610251849185115840731512, 4.68417151196206501694114695562, 5.69510265828470869796838823837, 6.89444137509146616852745549600, 8.065877480422518827358619720284, 8.775689139800608709658075136895, 9.906001270915681546609300347089, 10.83210992893769142493137773987

Graph of the ZZ-function along the critical line