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Properties

Label 2-315-1.1-c9-0-88
Degree 22
Conductor 315315
Sign −1-1
Analytic cond. 162.236162.236
Root an. cond. 12.737212.7372
Motivic weight 99
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 42.0·2-s + 1.25e3·4-s − 625·5-s + 2.40e3·7-s + 3.13e4·8-s − 2.62e4·10-s + 746.·11-s − 1.39e5·13-s + 1.01e5·14-s + 6.76e5·16-s − 6.57e5·17-s + 5.86e4·19-s − 7.86e5·20-s + 3.13e4·22-s − 2.05e6·23-s + 3.90e5·25-s − 5.85e6·26-s + 3.02e6·28-s + 2.78e6·29-s − 4.13e6·31-s + 1.23e7·32-s − 2.76e7·34-s − 1.50e6·35-s − 3.47e6·37-s + 2.46e6·38-s − 1.96e7·40-s − 6.46e6·41-s + ⋯
L(s)  = 1  + 1.85·2-s + 2.45·4-s − 0.447·5-s + 0.377·7-s + 2.70·8-s − 0.831·10-s + 0.0153·11-s − 1.35·13-s + 0.702·14-s + 2.58·16-s − 1.91·17-s + 0.103·19-s − 1.09·20-s + 0.0285·22-s − 1.53·23-s + 0.200·25-s − 2.51·26-s + 0.928·28-s + 0.730·29-s − 0.804·31-s + 2.08·32-s − 3.55·34-s − 0.169·35-s − 0.304·37-s + 0.192·38-s − 1.21·40-s − 0.357·41-s + ⋯

Functional equation

Λ(s)=(315s/2 ΓC(s) L(s)=(− Λ(10−s)\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}
Λ(s)=(315s/2 ΓC(s+9/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 315315    =    32⋅5⋅73^{2} \cdot 5 \cdot 7
Sign: −1-1
Analytic conductor: 162.236162.236
Root analytic conductor: 12.737212.7372
Motivic weight: 99
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 315, ( :9/2), −1)(2,\ 315,\ (\ :9/2),\ -1)

Particular Values

L(5)L(5) == 00
L(12)L(\frac12) == 00
L(112)L(\frac{11}{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)
bad3 1 1
5 1+625T 1 + 625T
7 1−2.40e3T 1 - 2.40e3T
good2 1−42.0T+512T2 1 - 42.0T + 512T^{2}
11 1−746.T+2.35e9T2 1 - 746.T + 2.35e9T^{2}
13 1+1.39e5T+1.06e10T2 1 + 1.39e5T + 1.06e10T^{2}
17 1+6.57e5T+1.18e11T2 1 + 6.57e5T + 1.18e11T^{2}
19 1−5.86e4T+3.22e11T2 1 - 5.86e4T + 3.22e11T^{2}
23 1+2.05e6T+1.80e12T2 1 + 2.05e6T + 1.80e12T^{2}
29 1−2.78e6T+1.45e13T2 1 - 2.78e6T + 1.45e13T^{2}
31 1+4.13e6T+2.64e13T2 1 + 4.13e6T + 2.64e13T^{2}
37 1+3.47e6T+1.29e14T2 1 + 3.47e6T + 1.29e14T^{2}
41 1+6.46e6T+3.27e14T2 1 + 6.46e6T + 3.27e14T^{2}
43 1+1.69e6T+5.02e14T2 1 + 1.69e6T + 5.02e14T^{2}
47 1+5.51e6T+1.11e15T2 1 + 5.51e6T + 1.11e15T^{2}
53 1+3.00e7T+3.29e15T2 1 + 3.00e7T + 3.29e15T^{2}
59 1−3.57e7T+8.66e15T2 1 - 3.57e7T + 8.66e15T^{2}
61 1−1.95e8T+1.16e16T2 1 - 1.95e8T + 1.16e16T^{2}
67 1−2.00e8T+2.72e16T2 1 - 2.00e8T + 2.72e16T^{2}
71 1−2.54e8T+4.58e16T2 1 - 2.54e8T + 4.58e16T^{2}
73 1+4.48e8T+5.88e16T2 1 + 4.48e8T + 5.88e16T^{2}
79 1+1.37e8T+1.19e17T2 1 + 1.37e8T + 1.19e17T^{2}
83 1+6.74e8T+1.86e17T2 1 + 6.74e8T + 1.86e17T^{2}
89 1+5.23e8T+3.50e17T2 1 + 5.23e8T + 3.50e17T^{2}
97 1+1.00e9T+7.60e17T2 1 + 1.00e9T + 7.60e17T^{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

−9.967077926009845161391535666422, −8.401279308230185668311816365483, −7.27394267319601216437375937907, −6.59248644576821164699407024647, −5.41881228336484319106344835026, −4.57829622528875754637731116254, −3.92304727808932936453962383673, −2.62973776203302289034250603237, −1.90139302121829854285751001864, 0, 1.90139302121829854285751001864, 2.62973776203302289034250603237, 3.92304727808932936453962383673, 4.57829622528875754637731116254, 5.41881228336484319106344835026, 6.59248644576821164699407024647, 7.27394267319601216437375937907, 8.401279308230185668311816365483, 9.967077926009845161391535666422

Graph of the ZZ-function along the critical line