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Properties

Label 2-312-1.1-c3-0-13
Degree 22
Conductor 312312
Sign −1-1
Analytic cond. 18.408518.4085
Root an. cond. 4.290524.29052
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  − 3·3-s + 13.6·5-s − 15.6·7-s + 9·9-s − 50.5·11-s + 13·13-s − 40.8·15-s + 2·17-s + 18.1·19-s + 46.8·21-s − 64·23-s + 60.7·25-s − 27·27-s − 103.·29-s + 34.4·31-s + 151.·33-s − 213.·35-s − 267.·37-s − 39·39-s − 147.·41-s − 166.·43-s + 122.·45-s − 325.·47-s − 98.6·49-s − 6·51-s − 111.·53-s − 688.·55-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.21·5-s − 0.843·7-s + 0.333·9-s − 1.38·11-s + 0.277·13-s − 0.703·15-s + 0.0285·17-s + 0.219·19-s + 0.487·21-s − 0.580·23-s + 0.486·25-s − 0.192·27-s − 0.664·29-s + 0.199·31-s + 0.799·33-s − 1.02·35-s − 1.18·37-s − 0.160·39-s − 0.561·41-s − 0.592·43-s + 0.406·45-s − 1.01·47-s − 0.287·49-s − 0.0164·51-s − 0.287·53-s − 1.68·55-s + ⋯

Functional equation

Λ(s)=(312s/2 ΓC(s) L(s)=(− Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}
Λ(s)=(312s/2 ΓC(s+3/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 312312    =    23⋅3⋅132^{3} \cdot 3 \cdot 13
Sign: −1-1
Analytic conductor: 18.408518.4085
Root analytic conductor: 4.290524.29052
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 312, ( :3/2), −1)(2,\ 312,\ (\ :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+3T 1 + 3T
13 1−13T 1 - 13T
good5 1−13.6T+125T2 1 - 13.6T + 125T^{2}
7 1+15.6T+343T2 1 + 15.6T + 343T^{2}
11 1+50.5T+1.33e3T2 1 + 50.5T + 1.33e3T^{2}
17 1−2T+4.91e3T2 1 - 2T + 4.91e3T^{2}
19 1−18.1T+6.85e3T2 1 - 18.1T + 6.85e3T^{2}
23 1+64T+1.21e4T2 1 + 64T + 1.21e4T^{2}
29 1+103.T+2.43e4T2 1 + 103.T + 2.43e4T^{2}
31 1−34.4T+2.97e4T2 1 - 34.4T + 2.97e4T^{2}
37 1+267.T+5.06e4T2 1 + 267.T + 5.06e4T^{2}
41 1+147.T+6.89e4T2 1 + 147.T + 6.89e4T^{2}
43 1+166.T+7.95e4T2 1 + 166.T + 7.95e4T^{2}
47 1+325.T+1.03e5T2 1 + 325.T + 1.03e5T^{2}
53 1+111.T+1.48e5T2 1 + 111.T + 1.48e5T^{2}
59 1−24.3T+2.05e5T2 1 - 24.3T + 2.05e5T^{2}
61 1+640.T+2.26e5T2 1 + 640.T + 2.26e5T^{2}
67 1+382.T+3.00e5T2 1 + 382.T + 3.00e5T^{2}
71 1−510.T+3.57e5T2 1 - 510.T + 3.57e5T^{2}
73 1+291.T+3.89e5T2 1 + 291.T + 3.89e5T^{2}
79 1+794.T+4.93e5T2 1 + 794.T + 4.93e5T^{2}
83 1−945.T+5.71e5T2 1 - 945.T + 5.71e5T^{2}
89 1−317.T+7.04e5T2 1 - 317.T + 7.04e5T^{2}
97 1−1.57e3T+9.12e5T2 1 - 1.57e3T + 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

−10.47408302912721709671895371089, −10.07219692754830156360763871506, −9.130084620197273060275968584433, −7.81594666308229760399335441251, −6.59181716711030375087842728346, −5.81576137688952866659119713553, −4.98556245367841609802608988824, −3.24264890822557178494249281106, −1.87730823113406259505474264815, 0, 1.87730823113406259505474264815, 3.24264890822557178494249281106, 4.98556245367841609802608988824, 5.81576137688952866659119713553, 6.59181716711030375087842728346, 7.81594666308229760399335441251, 9.130084620197273060275968584433, 10.07219692754830156360763871506, 10.47408302912721709671895371089

Graph of the ZZ-function along the critical line