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Properties

Label 2-312-1.1-c3-0-4
Degree 22
Conductor 312312
Sign 11
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 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·3-s − 7.11·5-s + 8.88·7-s + 9·9-s + 26·11-s − 13·13-s − 21.3·15-s + 16.2·17-s + 35.5·19-s + 26.6·21-s + 153.·23-s − 74.3·25-s + 27·27-s + 223.·29-s + 126.·31-s + 78·33-s − 63.2·35-s + 217.·37-s − 39·39-s + 105.·41-s − 183.·43-s − 64.0·45-s + 96.6·47-s − 264.·49-s + 48.6·51-s + 386.·53-s − 184.·55-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.636·5-s + 0.479·7-s + 0.333·9-s + 0.712·11-s − 0.277·13-s − 0.367·15-s + 0.231·17-s + 0.429·19-s + 0.276·21-s + 1.39·23-s − 0.595·25-s + 0.192·27-s + 1.43·29-s + 0.734·31-s + 0.411·33-s − 0.305·35-s + 0.966·37-s − 0.160·39-s + 0.403·41-s − 0.651·43-s − 0.212·45-s + 0.300·47-s − 0.769·49-s + 0.133·51-s + 1.00·53-s − 0.453·55-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: 11
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: 00
Selberg data: (2, 312, ( :3/2), 1)(2,\ 312,\ (\ :3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 2.2833552062.283355206
L(12)L(\frac12) ≈\approx 2.2833552062.283355206
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+7.11T+125T2 1 + 7.11T + 125T^{2}
7 1−8.88T+343T2 1 - 8.88T + 343T^{2}
11 1−26T+1.33e3T2 1 - 26T + 1.33e3T^{2}
17 1−16.2T+4.91e3T2 1 - 16.2T + 4.91e3T^{2}
19 1−35.5T+6.85e3T2 1 - 35.5T + 6.85e3T^{2}
23 1−153.T+1.21e4T2 1 - 153.T + 1.21e4T^{2}
29 1−223.T+2.43e4T2 1 - 223.T + 2.43e4T^{2}
31 1−126.T+2.97e4T2 1 - 126.T + 2.97e4T^{2}
37 1−217.T+5.06e4T2 1 - 217.T + 5.06e4T^{2}
41 1−105.T+6.89e4T2 1 - 105.T + 6.89e4T^{2}
43 1+183.T+7.95e4T2 1 + 183.T + 7.95e4T^{2}
47 1−96.6T+1.03e5T2 1 - 96.6T + 1.03e5T^{2}
53 1−386.T+1.48e5T2 1 - 386.T + 1.48e5T^{2}
59 1−34.5T+2.05e5T2 1 - 34.5T + 2.05e5T^{2}
61 1−274.T+2.26e5T2 1 - 274.T + 2.26e5T^{2}
67 1−93.9T+3.00e5T2 1 - 93.9T + 3.00e5T^{2}
71 1+741.T+3.57e5T2 1 + 741.T + 3.57e5T^{2}
73 1−640.T+3.89e5T2 1 - 640.T + 3.89e5T^{2}
79 1+182.T+4.93e5T2 1 + 182.T + 4.93e5T^{2}
83 1+288.T+5.71e5T2 1 + 288.T + 5.71e5T^{2}
89 1−963.T+7.04e5T2 1 - 963.T + 7.04e5T^{2}
97 1+481.T+9.12e5T2 1 + 481.T + 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

−11.42214398917128688713621417985, −10.24788617272732909805315794294, −9.264813226358060540585988578082, −8.346252110582788825116967335954, −7.54957521691544461779378250866, −6.53715315204666169112077639816, −5.00426301943975155280014032240, −3.98523586614477440481232678366, −2.76919953794383634385098601660, −1.10022464665493037666502970190, 1.10022464665493037666502970190, 2.76919953794383634385098601660, 3.98523586614477440481232678366, 5.00426301943975155280014032240, 6.53715315204666169112077639816, 7.54957521691544461779378250866, 8.346252110582788825116967335954, 9.264813226358060540585988578082, 10.24788617272732909805315794294, 11.42214398917128688713621417985

Graph of the ZZ-function along the critical line