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Properties

Label 2-1275-1.1-c3-0-61
Degree 22
Conductor 12751275
Sign 11
Analytic cond. 75.227475.2274
Root an. cond. 8.673378.67337
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  − 4.24·2-s + 3·3-s + 9.99·4-s − 12.7·6-s + 20.9·7-s − 8.48·8-s + 9·9-s + 16.0·11-s + 29.9·12-s + 34.9·13-s − 88.9·14-s − 44.0·16-s + 17·17-s − 38.1·18-s − 80.8·19-s + 62.9·21-s − 68.0·22-s + 115.·23-s − 25.4·24-s − 148.·26-s + 27·27-s + 209.·28-s + 154.·29-s + 299.·31-s + 254.·32-s + 48.0·33-s − 72.1·34-s + ⋯
L(s)  = 1  − 1.49·2-s + 0.577·3-s + 1.24·4-s − 0.866·6-s + 1.13·7-s − 0.374·8-s + 0.333·9-s + 0.439·11-s + 0.721·12-s + 0.745·13-s − 1.69·14-s − 0.687·16-s + 0.242·17-s − 0.500·18-s − 0.976·19-s + 0.653·21-s − 0.659·22-s + 1.05·23-s − 0.216·24-s − 1.11·26-s + 0.192·27-s + 1.41·28-s + 0.986·29-s + 1.73·31-s + 1.40·32-s + 0.253·33-s − 0.363·34-s + ⋯

Functional equation

Λ(s)=(1275s/2 ΓC(s) L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 1275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(1275s/2 ΓC(s+3/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1275 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 12751275    =    3⋅52⋅173 \cdot 5^{2} \cdot 17
Sign: 11
Analytic conductor: 75.227475.2274
Root analytic conductor: 8.673378.67337
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 1275, ( :3/2), 1)(2,\ 1275,\ (\ :3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 1.6786415531.678641553
L(12)L(\frac12) ≈\approx 1.6786415531.678641553
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)
bad3 1−3T 1 - 3T
5 1 1
17 1−17T 1 - 17T
good2 1+4.24T+8T2 1 + 4.24T + 8T^{2}
7 1−20.9T+343T2 1 - 20.9T + 343T^{2}
11 1−16.0T+1.33e3T2 1 - 16.0T + 1.33e3T^{2}
13 1−34.9T+2.19e3T2 1 - 34.9T + 2.19e3T^{2}
19 1+80.8T+6.85e3T2 1 + 80.8T + 6.85e3T^{2}
23 1−115.T+1.21e4T2 1 - 115.T + 1.21e4T^{2}
29 1−154.T+2.43e4T2 1 - 154.T + 2.43e4T^{2}
31 1−299.T+2.97e4T2 1 - 299.T + 2.97e4T^{2}
37 1+315.T+5.06e4T2 1 + 315.T + 5.06e4T^{2}
41 1−132.T+6.89e4T2 1 - 132.T + 6.89e4T^{2}
43 1−23.1T+7.95e4T2 1 - 23.1T + 7.95e4T^{2}
47 1+260.T+1.03e5T2 1 + 260.T + 1.03e5T^{2}
53 1−676.T+1.48e5T2 1 - 676.T + 1.48e5T^{2}
59 1−629.T+2.05e5T2 1 - 629.T + 2.05e5T^{2}
61 1+461.T+2.26e5T2 1 + 461.T + 2.26e5T^{2}
67 1−789.T+3.00e5T2 1 - 789.T + 3.00e5T^{2}
71 1+686.T+3.57e5T2 1 + 686.T + 3.57e5T^{2}
73 1+484.T+3.89e5T2 1 + 484.T + 3.89e5T^{2}
79 1−254T+4.93e5T2 1 - 254T + 4.93e5T^{2}
83 1+548.T+5.71e5T2 1 + 548.T + 5.71e5T^{2}
89 1−925.T+7.04e5T2 1 - 925.T + 7.04e5T^{2}
97 1+732.T+9.12e5T2 1 + 732.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

−9.003535998651876752019289985465, −8.516287773935220348945673348710, −8.107784427391948959391271198791, −7.12978735617992524184432187813, −6.41880477895644631301898358519, −4.98333828728322345248571852349, −4.08060461168740891044659634658, −2.68299052140633936988185510876, −1.62132933971087292184976279183, −0.879154950982805729397334529224, 0.879154950982805729397334529224, 1.62132933971087292184976279183, 2.68299052140633936988185510876, 4.08060461168740891044659634658, 4.98333828728322345248571852349, 6.41880477895644631301898358519, 7.12978735617992524184432187813, 8.107784427391948959391271198791, 8.516287773935220348945673348710, 9.003535998651876752019289985465

Graph of the ZZ-function along the critical line