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Properties

Label 2-312-1.1-c3-0-11
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 − 5.46·5-s + 12.3·7-s + 9·9-s − 6.78·11-s − 13·13-s + 16.3·15-s + 72.0·17-s − 99.2·19-s − 37.1·21-s + 120.·23-s − 95.1·25-s − 27·27-s − 185.·29-s − 85.4·31-s + 20.3·33-s − 67.7·35-s − 340.·37-s + 39·39-s − 427.·41-s + 64.9·43-s − 49.1·45-s − 39.2·47-s − 189.·49-s − 216.·51-s − 21.4·53-s + 37.0·55-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.488·5-s + 0.669·7-s + 0.333·9-s − 0.185·11-s − 0.277·13-s + 0.282·15-s + 1.02·17-s − 1.19·19-s − 0.386·21-s + 1.09·23-s − 0.761·25-s − 0.192·27-s − 1.18·29-s − 0.495·31-s + 0.107·33-s − 0.327·35-s − 1.51·37-s + 0.160·39-s − 1.62·41-s + 0.230·43-s − 0.162·45-s − 0.121·47-s − 0.552·49-s − 0.593·51-s − 0.0555·53-s + 0.0908·55-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+5.46T+125T2 1 + 5.46T + 125T^{2}
7 1−12.3T+343T2 1 - 12.3T + 343T^{2}
11 1+6.78T+1.33e3T2 1 + 6.78T + 1.33e3T^{2}
17 1−72.0T+4.91e3T2 1 - 72.0T + 4.91e3T^{2}
19 1+99.2T+6.85e3T2 1 + 99.2T + 6.85e3T^{2}
23 1−120.T+1.21e4T2 1 - 120.T + 1.21e4T^{2}
29 1+185.T+2.43e4T2 1 + 185.T + 2.43e4T^{2}
31 1+85.4T+2.97e4T2 1 + 85.4T + 2.97e4T^{2}
37 1+340.T+5.06e4T2 1 + 340.T + 5.06e4T^{2}
41 1+427.T+6.89e4T2 1 + 427.T + 6.89e4T^{2}
43 1−64.9T+7.95e4T2 1 - 64.9T + 7.95e4T^{2}
47 1+39.2T+1.03e5T2 1 + 39.2T + 1.03e5T^{2}
53 1+21.4T+1.48e5T2 1 + 21.4T + 1.48e5T^{2}
59 1−62.4T+2.05e5T2 1 - 62.4T + 2.05e5T^{2}
61 1+423.T+2.26e5T2 1 + 423.T + 2.26e5T^{2}
67 1−451.T+3.00e5T2 1 - 451.T + 3.00e5T^{2}
71 1−335.T+3.57e5T2 1 - 335.T + 3.57e5T^{2}
73 1+1.01e3T+3.89e5T2 1 + 1.01e3T + 3.89e5T^{2}
79 1+398.T+4.93e5T2 1 + 398.T + 4.93e5T^{2}
83 1+865.T+5.71e5T2 1 + 865.T + 5.71e5T^{2}
89 1−641.T+7.04e5T2 1 - 641.T + 7.04e5T^{2}
97 1−1.38e3T+9.12e5T2 1 - 1.38e3T + 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

−10.90042318070465831834560823108, −10.04122213100449052792144620156, −8.798266037766021627463178879892, −7.81676760546915516721479235184, −6.95895285607915455130138091454, −5.64847200944659758610050754833, −4.76482157037134973651968638957, −3.52779358656910079028742869930, −1.72098342268013594403718883638, 0, 1.72098342268013594403718883638, 3.52779358656910079028742869930, 4.76482157037134973651968638957, 5.64847200944659758610050754833, 6.95895285607915455130138091454, 7.81676760546915516721479235184, 8.798266037766021627463178879892, 10.04122213100449052792144620156, 10.90042318070465831834560823108

Graph of the ZZ-function along the critical line