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Properties

Label 2-312-1.1-c3-0-9
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 − 7.63·5-s + 5.63·7-s + 9·9-s + 34.5·11-s + 13·13-s + 22.8·15-s + 2·17-s − 88.1·19-s − 16.8·21-s − 64·23-s − 66.7·25-s − 27·27-s + 23.7·29-s − 284.·31-s − 103.·33-s − 42.9·35-s + 115.·37-s − 39·39-s + 1.41·41-s − 337.·43-s − 68.6·45-s − 198.·47-s − 311.·49-s − 6·51-s + 59.0·53-s − 263.·55-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.682·5-s + 0.303·7-s + 0.333·9-s + 0.946·11-s + 0.277·13-s + 0.394·15-s + 0.0285·17-s − 1.06·19-s − 0.175·21-s − 0.580·23-s − 0.534·25-s − 0.192·27-s + 0.152·29-s − 1.64·31-s − 0.546·33-s − 0.207·35-s + 0.512·37-s − 0.160·39-s + 0.00537·41-s − 1.19·43-s − 0.227·45-s − 0.615·47-s − 0.907·49-s − 0.0164·51-s + 0.153·53-s − 0.645·55-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+7.63T+125T2 1 + 7.63T + 125T^{2}
7 1−5.63T+343T2 1 - 5.63T + 343T^{2}
11 1−34.5T+1.33e3T2 1 - 34.5T + 1.33e3T^{2}
17 1−2T+4.91e3T2 1 - 2T + 4.91e3T^{2}
19 1+88.1T+6.85e3T2 1 + 88.1T + 6.85e3T^{2}
23 1+64T+1.21e4T2 1 + 64T + 1.21e4T^{2}
29 1−23.7T+2.43e4T2 1 - 23.7T + 2.43e4T^{2}
31 1+284.T+2.97e4T2 1 + 284.T + 2.97e4T^{2}
37 1−115.T+5.06e4T2 1 - 115.T + 5.06e4T^{2}
41 1−1.41T+6.89e4T2 1 - 1.41T + 6.89e4T^{2}
43 1+337.T+7.95e4T2 1 + 337.T + 7.95e4T^{2}
47 1+198.T+1.03e5T2 1 + 198.T + 1.03e5T^{2}
53 1−59.0T+1.48e5T2 1 - 59.0T + 1.48e5T^{2}
59 1+188.T+2.05e5T2 1 + 188.T + 2.05e5T^{2}
61 1−336.T+2.26e5T2 1 - 336.T + 2.26e5T^{2}
67 1+531.T+3.00e5T2 1 + 531.T + 3.00e5T^{2}
71 1+510.T+3.57e5T2 1 + 510.T + 3.57e5T^{2}
73 1+164.T+3.89e5T2 1 + 164.T + 3.89e5T^{2}
79 1+29.3T+4.93e5T2 1 + 29.3T + 4.93e5T^{2}
83 1+117.T+5.71e5T2 1 + 117.T + 5.71e5T^{2}
89 1−508.T+7.04e5T2 1 - 508.T + 7.04e5T^{2}
97 1+1.02e3T+9.12e5T2 1 + 1.02e3T + 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

−11.02978616120094852203988797375, −9.940840277987285178439966104772, −8.825879345907695581214683451340, −7.87759948175251218017594841934, −6.80782589689300687629677522991, −5.86677578586460800073622475365, −4.54013886905400691343241210497, −3.65589810597696841341480919043, −1.68543225700646612433393461335, 0, 1.68543225700646612433393461335, 3.65589810597696841341480919043, 4.54013886905400691343241210497, 5.86677578586460800073622475365, 6.80782589689300687629677522991, 7.87759948175251218017594841934, 8.825879345907695581214683451340, 9.940840277987285178439966104772, 11.02978616120094852203988797375

Graph of the ZZ-function along the critical line