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Properties

Label 2-312-1.1-c3-0-2
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 − 20.1·5-s − 19.6·7-s + 9·9-s + 13.5·11-s + 13·13-s − 60.4·15-s + 116.·17-s + 68.1·19-s − 59.0·21-s + 122.·23-s + 280.·25-s + 27·27-s + 204.·29-s − 194.·31-s + 40.6·33-s + 396.·35-s − 142.·37-s + 39·39-s − 175.·41-s − 219.·43-s − 181.·45-s + 236.·47-s + 45.0·49-s + 349.·51-s − 628.·53-s − 273.·55-s + ⋯
L(s)  = 1  + 0.577·3-s − 1.80·5-s − 1.06·7-s + 0.333·9-s + 0.371·11-s + 0.277·13-s − 1.04·15-s + 1.66·17-s + 0.822·19-s − 0.614·21-s + 1.10·23-s + 2.24·25-s + 0.192·27-s + 1.30·29-s − 1.12·31-s + 0.214·33-s + 1.91·35-s − 0.634·37-s + 0.160·39-s − 0.666·41-s − 0.778·43-s − 0.600·45-s + 0.733·47-s + 0.131·49-s + 0.958·51-s − 1.62·53-s − 0.669·55-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 1.4551504051.455150405
L(12)L(\frac12) ≈\approx 1.4551504051.455150405
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+20.1T+125T2 1 + 20.1T + 125T^{2}
7 1+19.6T+343T2 1 + 19.6T + 343T^{2}
11 1−13.5T+1.33e3T2 1 - 13.5T + 1.33e3T^{2}
17 1−116.T+4.91e3T2 1 - 116.T + 4.91e3T^{2}
19 1−68.1T+6.85e3T2 1 - 68.1T + 6.85e3T^{2}
23 1−122.T+1.21e4T2 1 - 122.T + 1.21e4T^{2}
29 1−204.T+2.43e4T2 1 - 204.T + 2.43e4T^{2}
31 1+194.T+2.97e4T2 1 + 194.T + 2.97e4T^{2}
37 1+142.T+5.06e4T2 1 + 142.T + 5.06e4T^{2}
41 1+175.T+6.89e4T2 1 + 175.T + 6.89e4T^{2}
43 1+219.T+7.95e4T2 1 + 219.T + 7.95e4T^{2}
47 1−236.T+1.03e5T2 1 - 236.T + 1.03e5T^{2}
53 1+628.T+1.48e5T2 1 + 628.T + 1.48e5T^{2}
59 1−446.T+2.05e5T2 1 - 446.T + 2.05e5T^{2}
61 1−224.T+2.26e5T2 1 - 224.T + 2.26e5T^{2}
67 1−165.T+3.00e5T2 1 - 165.T + 3.00e5T^{2}
71 1−902.T+3.57e5T2 1 - 902.T + 3.57e5T^{2}
73 1+15.1T+3.89e5T2 1 + 15.1T + 3.89e5T^{2}
79 1−670.T+4.93e5T2 1 - 670.T + 4.93e5T^{2}
83 1−1.04e3T+5.71e5T2 1 - 1.04e3T + 5.71e5T^{2}
89 1+562.T+7.04e5T2 1 + 562.T + 7.04e5T^{2}
97 1−1.64e3T+9.12e5T2 1 - 1.64e3T + 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

−11.39429318280176290491104612639, −10.26319618218411838502990788453, −9.261487560716465069374359732782, −8.305392067869199820829314185354, −7.48283110829867664917863584212, −6.67751471889653331595213889986, −5.02071010652252472414498437730, −3.52634100507530736562074259558, −3.30149869625459484545763770135, −0.821943270193890418587602049894, 0.821943270193890418587602049894, 3.30149869625459484545763770135, 3.52634100507530736562074259558, 5.02071010652252472414498437730, 6.67751471889653331595213889986, 7.48283110829867664917863584212, 8.305392067869199820829314185354, 9.261487560716465069374359732782, 10.26319618218411838502990788453, 11.39429318280176290491104612639

Graph of the ZZ-function along the critical line