↑

Properties

Label 2-315-1.1-c9-0-58
Degree 22
Conductor 315315
Sign 11
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 00

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 38.8·2-s + 996.·4-s + 625·5-s + 2.40e3·7-s + 1.88e4·8-s + 2.42e4·10-s − 6.97e3·11-s + 4.56e4·13-s + 9.32e4·14-s + 2.20e5·16-s + 1.54e5·17-s − 3.80e5·19-s + 6.22e5·20-s − 2.70e5·22-s + 1.66e6·23-s + 3.90e5·25-s + 1.77e6·26-s + 2.39e6·28-s + 2.23e6·29-s + 5.92e6·31-s − 1.06e6·32-s + 6.01e6·34-s + 1.50e6·35-s + 4.08e6·37-s − 1.47e7·38-s + 1.17e7·40-s − 1.62e6·41-s + ⋯
L(s)  = 1  + 1.71·2-s + 1.94·4-s + 0.447·5-s + 0.377·7-s + 1.62·8-s + 0.767·10-s − 0.143·11-s + 0.442·13-s + 0.648·14-s + 0.841·16-s + 0.449·17-s − 0.670·19-s + 0.870·20-s − 0.246·22-s + 1.24·23-s + 0.200·25-s + 0.760·26-s + 0.735·28-s + 0.586·29-s + 1.15·31-s − 0.180·32-s + 0.771·34-s + 0.169·35-s + 0.358·37-s − 1.15·38-s + 0.726·40-s − 0.0895·41-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: 11
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: 00
Selberg data: (2, 315, ( :9/2), 1)(2,\ 315,\ (\ :9/2),\ 1)

Particular Values

L(5)L(5) ≈\approx 8.8070675238.807067523
L(12)L(\frac12) ≈\approx 8.8070675238.807067523
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 1−38.8T+512T2 1 - 38.8T + 512T^{2}
11 1+6.97e3T+2.35e9T2 1 + 6.97e3T + 2.35e9T^{2}
13 1−4.56e4T+1.06e10T2 1 - 4.56e4T + 1.06e10T^{2}
17 1−1.54e5T+1.18e11T2 1 - 1.54e5T + 1.18e11T^{2}
19 1+3.80e5T+3.22e11T2 1 + 3.80e5T + 3.22e11T^{2}
23 1−1.66e6T+1.80e12T2 1 - 1.66e6T + 1.80e12T^{2}
29 1−2.23e6T+1.45e13T2 1 - 2.23e6T + 1.45e13T^{2}
31 1−5.92e6T+2.64e13T2 1 - 5.92e6T + 2.64e13T^{2}
37 1−4.08e6T+1.29e14T2 1 - 4.08e6T + 1.29e14T^{2}
41 1+1.62e6T+3.27e14T2 1 + 1.62e6T + 3.27e14T^{2}
43 1−2.61e7T+5.02e14T2 1 - 2.61e7T + 5.02e14T^{2}
47 1−3.12e7T+1.11e15T2 1 - 3.12e7T + 1.11e15T^{2}
53 1+8.31e7T+3.29e15T2 1 + 8.31e7T + 3.29e15T^{2}
59 1−3.03e7T+8.66e15T2 1 - 3.03e7T + 8.66e15T^{2}
61 1+1.10e8T+1.16e16T2 1 + 1.10e8T + 1.16e16T^{2}
67 1+1.01e8T+2.72e16T2 1 + 1.01e8T + 2.72e16T^{2}
71 1−3.91e8T+4.58e16T2 1 - 3.91e8T + 4.58e16T^{2}
73 1−2.16e8T+5.88e16T2 1 - 2.16e8T + 5.88e16T^{2}
79 1−1.00e8T+1.19e17T2 1 - 1.00e8T + 1.19e17T^{2}
83 1−3.07e8T+1.86e17T2 1 - 3.07e8T + 1.86e17T^{2}
89 1+2.71e8T+3.50e17T2 1 + 2.71e8T + 3.50e17T^{2}
97 1+2.51e8T+7.60e17T2 1 + 2.51e8T + 7.60e17T^{2}
show more
show less
   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

−10.54467868267063436405676907601, −9.195120008723961496640752713863, −7.965626980883915701642420089055, −6.77343651103215939858800134196, −6.01668921505784330015371331683, −5.08668982455218909441288772527, −4.31136069549469982418368994431, −3.16210359405961876123728902440, −2.28577097016992627238319050224, −1.05249568027173207308618487084, 1.05249568027173207308618487084, 2.28577097016992627238319050224, 3.16210359405961876123728902440, 4.31136069549469982418368994431, 5.08668982455218909441288772527, 6.01668921505784330015371331683, 6.77343651103215939858800134196, 7.965626980883915701642420089055, 9.195120008723961496640752713863, 10.54467868267063436405676907601

Graph of the ZZ-function along the critical line