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Properties

Label 2-312-1.1-c3-0-1
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 − 16.8·5-s − 4.83·7-s + 9·9-s − 39.6·11-s − 13·13-s + 50.4·15-s − 4.33·17-s + 28.8·19-s + 14.4·21-s + 0.670·23-s + 158.·25-s − 27·27-s − 6·29-s + 274.·31-s + 118.·33-s + 81.3·35-s + 84.3·37-s + 39·39-s + 209.·41-s + 364.·43-s − 151.·45-s + 239.·47-s − 319.·49-s + 13.0·51-s − 415.·53-s + 667.·55-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.50·5-s − 0.260·7-s + 0.333·9-s − 1.08·11-s − 0.277·13-s + 0.869·15-s − 0.0618·17-s + 0.348·19-s + 0.150·21-s + 0.00607·23-s + 1.26·25-s − 0.192·27-s − 0.0384·29-s + 1.59·31-s + 0.627·33-s + 0.392·35-s + 0.374·37-s + 0.160·39-s + 0.796·41-s + 1.29·43-s − 0.501·45-s + 0.742·47-s − 0.931·49-s + 0.0357·51-s − 1.07·53-s + 1.63·55-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 0.71771206560.7177120656
L(12)L(\frac12) ≈\approx 0.71771206560.7177120656
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+16.8T+125T2 1 + 16.8T + 125T^{2}
7 1+4.83T+343T2 1 + 4.83T + 343T^{2}
11 1+39.6T+1.33e3T2 1 + 39.6T + 1.33e3T^{2}
17 1+4.33T+4.91e3T2 1 + 4.33T + 4.91e3T^{2}
19 1−28.8T+6.85e3T2 1 - 28.8T + 6.85e3T^{2}
23 1−0.670T+1.21e4T2 1 - 0.670T + 1.21e4T^{2}
29 1+6T+2.43e4T2 1 + 6T + 2.43e4T^{2}
31 1−274.T+2.97e4T2 1 - 274.T + 2.97e4T^{2}
37 1−84.3T+5.06e4T2 1 - 84.3T + 5.06e4T^{2}
41 1−209.T+6.89e4T2 1 - 209.T + 6.89e4T^{2}
43 1−364.T+7.95e4T2 1 - 364.T + 7.95e4T^{2}
47 1−239.T+1.03e5T2 1 - 239.T + 1.03e5T^{2}
53 1+415.T+1.48e5T2 1 + 415.T + 1.48e5T^{2}
59 1−78T+2.05e5T2 1 - 78T + 2.05e5T^{2}
61 1+813.T+2.26e5T2 1 + 813.T + 2.26e5T^{2}
67 1−858.T+3.00e5T2 1 - 858.T + 3.00e5T^{2}
71 1+213.T+3.57e5T2 1 + 213.T + 3.57e5T^{2}
73 1−568.T+3.89e5T2 1 - 568.T + 3.89e5T^{2}
79 1−583.T+4.93e5T2 1 - 583.T + 4.93e5T^{2}
83 1−162.T+5.71e5T2 1 - 162.T + 5.71e5T^{2}
89 1−1.41e3T+7.04e5T2 1 - 1.41e3T + 7.04e5T^{2}
97 1+798.T+9.12e5T2 1 + 798.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

−11.25436728830919157413193400354, −10.57088422815317014673382692516, −9.442446914904792999766688582196, −8.062716069301654496009204325083, −7.57122280748334288161192085717, −6.39678236371818131417447254622, −5.08873577320498880915398998606, −4.14506193888781918586256226119, −2.84525773573596656463848730287, −0.58206734096837513085031327682, 0.58206734096837513085031327682, 2.84525773573596656463848730287, 4.14506193888781918586256226119, 5.08873577320498880915398998606, 6.39678236371818131417447254622, 7.57122280748334288161192085717, 8.062716069301654496009204325083, 9.442446914904792999766688582196, 10.57088422815317014673382692516, 11.25436728830919157413193400354

Graph of the ZZ-function along the critical line