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Properties

Label 2-310-1.1-c7-0-35
Degree 22
Conductor 310310
Sign −1-1
Analytic cond. 96.839396.8393
Root an. cond. 9.840699.84069
Motivic weight 77
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 8·2-s − 53.1·3-s + 64·4-s − 125·5-s + 425.·6-s + 1.53e3·7-s − 512·8-s + 641.·9-s + 1.00e3·10-s − 5.52e3·11-s − 3.40e3·12-s + 1.33e4·13-s − 1.23e4·14-s + 6.64e3·15-s + 4.09e3·16-s − 1.29e4·17-s − 5.12e3·18-s − 3.25e4·19-s − 8.00e3·20-s − 8.18e4·21-s + 4.41e4·22-s − 1.58e4·23-s + 2.72e4·24-s + 1.56e4·25-s − 1.07e5·26-s + 8.22e4·27-s + 9.85e4·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.13·3-s + 0.5·4-s − 0.447·5-s + 0.804·6-s + 1.69·7-s − 0.353·8-s + 0.293·9-s + 0.316·10-s − 1.25·11-s − 0.568·12-s + 1.68·13-s − 1.19·14-s + 0.508·15-s + 0.250·16-s − 0.641·17-s − 0.207·18-s − 1.08·19-s − 0.223·20-s − 1.92·21-s + 0.884·22-s − 0.271·23-s + 0.402·24-s + 0.199·25-s − 1.19·26-s + 0.803·27-s + 0.848·28-s + ⋯

Functional equation

Λ(s)=(310s/2 ΓC(s) L(s)=(− Λ(8−s)\begin{aligned}\Lambda(s)=\mathstrut & 310 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}
Λ(s)=(310s/2 ΓC(s+7/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 310 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 310310    =    2⋅5⋅312 \cdot 5 \cdot 31
Sign: −1-1
Analytic conductor: 96.839396.8393
Root analytic conductor: 9.840699.84069
Motivic weight: 77
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 310, ( :7/2), −1)(2,\ 310,\ (\ :7/2),\ -1)

Particular Values

L(4)L(4) == 00
L(12)L(\frac12) == 00
L(92)L(\frac{9}{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+8T 1 + 8T
5 1+125T 1 + 125T
31 1−2.97e4T 1 - 2.97e4T
good3 1+53.1T+2.18e3T2 1 + 53.1T + 2.18e3T^{2}
7 1−1.53e3T+8.23e5T2 1 - 1.53e3T + 8.23e5T^{2}
11 1+5.52e3T+1.94e7T2 1 + 5.52e3T + 1.94e7T^{2}
13 1−1.33e4T+6.27e7T2 1 - 1.33e4T + 6.27e7T^{2}
17 1+1.29e4T+4.10e8T2 1 + 1.29e4T + 4.10e8T^{2}
19 1+3.25e4T+8.93e8T2 1 + 3.25e4T + 8.93e8T^{2}
23 1+1.58e4T+3.40e9T2 1 + 1.58e4T + 3.40e9T^{2}
29 1+7.06e4T+1.72e10T2 1 + 7.06e4T + 1.72e10T^{2}
37 1+5.39e5T+9.49e10T2 1 + 5.39e5T + 9.49e10T^{2}
41 1−2.22e5T+1.94e11T2 1 - 2.22e5T + 1.94e11T^{2}
43 1−5.73e5T+2.71e11T2 1 - 5.73e5T + 2.71e11T^{2}
47 1−9.29e5T+5.06e11T2 1 - 9.29e5T + 5.06e11T^{2}
53 1−6.56e4T+1.17e12T2 1 - 6.56e4T + 1.17e12T^{2}
59 1−9.74e5T+2.48e12T2 1 - 9.74e5T + 2.48e12T^{2}
61 1−1.85e6T+3.14e12T2 1 - 1.85e6T + 3.14e12T^{2}
67 1−1.16e5T+6.06e12T2 1 - 1.16e5T + 6.06e12T^{2}
71 1+2.10e6T+9.09e12T2 1 + 2.10e6T + 9.09e12T^{2}
73 1−3.05e6T+1.10e13T2 1 - 3.05e6T + 1.10e13T^{2}
79 1+1.69e6T+1.92e13T2 1 + 1.69e6T + 1.92e13T^{2}
83 1+7.41e6T+2.71e13T2 1 + 7.41e6T + 2.71e13T^{2}
89 1−5.80e6T+4.42e13T2 1 - 5.80e6T + 4.42e13T^{2}
97 1−1.48e7T+8.07e13T2 1 - 1.48e7T + 8.07e13T^{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

−10.63863337392573531521837488156, −8.679996656666741462816159882719, −8.293660570777776339708564508287, −7.22515773722469750211319590445, −6.00740614946798795977013860763, −5.19885477851822604934628034207, −4.08862196297858400313116921841, −2.23274030983479114006613027380, −1.06699615105444387296562391823, 0, 1.06699615105444387296562391823, 2.23274030983479114006613027380, 4.08862196297858400313116921841, 5.19885477851822604934628034207, 6.00740614946798795977013860763, 7.22515773722469750211319590445, 8.293660570777776339708564508287, 8.679996656666741462816159882719, 10.63863337392573531521837488156

Graph of the ZZ-function along the critical line