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Properties

Label 2-18e2-1.1-c3-0-8
Degree 22
Conductor 324324
Sign −1-1
Analytic cond. 19.116619.1166
Root an. cond. 4.372254.37225
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4.89·5-s − 10.6·7-s + 35.4·11-s + 72.7·13-s − 127.·17-s − 46.3·19-s − 131.·23-s − 101.·25-s − 137.·29-s − 106.·31-s + 52.1·35-s + 137.·37-s − 71.7·41-s + 376.·43-s − 613.·47-s − 229.·49-s − 431.·53-s − 173.·55-s − 285.·59-s − 43.9·61-s − 356.·65-s − 45.2·67-s + 357.·71-s + 530.·73-s − 377.·77-s − 195.·79-s + 760.·83-s + ⋯
L(s)  = 1  − 0.438·5-s − 0.575·7-s + 0.971·11-s + 1.55·13-s − 1.81·17-s − 0.560·19-s − 1.18·23-s − 0.808·25-s − 0.880·29-s − 0.614·31-s + 0.252·35-s + 0.610·37-s − 0.273·41-s + 1.33·43-s − 1.90·47-s − 0.669·49-s − 1.11·53-s − 0.425·55-s − 0.630·59-s − 0.0922·61-s − 0.679·65-s − 0.0824·67-s + 0.597·71-s + 0.850·73-s − 0.559·77-s − 0.277·79-s + 1.00·83-s + ⋯

Functional equation

Λ(s)=(324s/2 ΓC(s) L(s)=(− Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}
Λ(s)=(324s/2 ΓC(s+3/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 324324    =    22⋅342^{2} \cdot 3^{4}
Sign: −1-1
Analytic conductor: 19.116619.1166
Root analytic conductor: 4.372254.37225
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 324, ( :3/2), −1)(2,\ 324,\ (\ :3/2),\ -1)

Particular Values

L(2)L(2) == 00
L(12)L(\frac12) == 00
L(52)L(\frac{5}{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
3 1 1
good5 1+4.89T+125T2 1 + 4.89T + 125T^{2}
7 1+10.6T+343T2 1 + 10.6T + 343T^{2}
11 1−35.4T+1.33e3T2 1 - 35.4T + 1.33e3T^{2}
13 1−72.7T+2.19e3T2 1 - 72.7T + 2.19e3T^{2}
17 1+127.T+4.91e3T2 1 + 127.T + 4.91e3T^{2}
19 1+46.3T+6.85e3T2 1 + 46.3T + 6.85e3T^{2}
23 1+131.T+1.21e4T2 1 + 131.T + 1.21e4T^{2}
29 1+137.T+2.43e4T2 1 + 137.T + 2.43e4T^{2}
31 1+106.T+2.97e4T2 1 + 106.T + 2.97e4T^{2}
37 1−137.T+5.06e4T2 1 - 137.T + 5.06e4T^{2}
41 1+71.7T+6.89e4T2 1 + 71.7T + 6.89e4T^{2}
43 1−376.T+7.95e4T2 1 - 376.T + 7.95e4T^{2}
47 1+613.T+1.03e5T2 1 + 613.T + 1.03e5T^{2}
53 1+431.T+1.48e5T2 1 + 431.T + 1.48e5T^{2}
59 1+285.T+2.05e5T2 1 + 285.T + 2.05e5T^{2}
61 1+43.9T+2.26e5T2 1 + 43.9T + 2.26e5T^{2}
67 1+45.2T+3.00e5T2 1 + 45.2T + 3.00e5T^{2}
71 1−357.T+3.57e5T2 1 - 357.T + 3.57e5T^{2}
73 1−530.T+3.89e5T2 1 - 530.T + 3.89e5T^{2}
79 1+195.T+4.93e5T2 1 + 195.T + 4.93e5T^{2}
83 1−760.T+5.71e5T2 1 - 760.T + 5.71e5T^{2}
89 1+1.21e3T+7.04e5T2 1 + 1.21e3T + 7.04e5T^{2}
97 1+1.10e3T+9.12e5T2 1 + 1.10e3T + 9.12e5T^{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

−11.02608164315251700458172723562, −9.640751371593460439974640225125, −8.872939536217413203682903037381, −7.955495241578396877950963273317, −6.59555773876972167411142405218, −6.07838104573034270751177817638, −4.30092263797932407365632121256, −3.59427183906464709241119251694, −1.82945016722240501372940533717, 0, 1.82945016722240501372940533717, 3.59427183906464709241119251694, 4.30092263797932407365632121256, 6.07838104573034270751177817638, 6.59555773876972167411142405218, 7.955495241578396877950963273317, 8.872939536217413203682903037381, 9.640751371593460439974640225125, 11.02608164315251700458172723562

Graph of the ZZ-function along the critical line