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Properties

Label 2-627-1.1-c3-0-6
Degree 22
Conductor 627627
Sign 11
Analytic cond. 36.994136.9941
Root an. cond. 6.082286.08228
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  − 2.70·2-s + 3·3-s − 0.678·4-s − 4.71·5-s − 8.11·6-s − 33.7·7-s + 23.4·8-s + 9·9-s + 12.7·10-s − 11·11-s − 2.03·12-s + 58.7·13-s + 91.2·14-s − 14.1·15-s − 58.1·16-s − 115.·17-s − 24.3·18-s + 19·19-s + 3.19·20-s − 101.·21-s + 29.7·22-s − 188.·23-s + 70.4·24-s − 102.·25-s − 158.·26-s + 27·27-s + 22.8·28-s + ⋯
L(s)  = 1  − 0.956·2-s + 0.577·3-s − 0.0847·4-s − 0.421·5-s − 0.552·6-s − 1.82·7-s + 1.03·8-s + 0.333·9-s + 0.403·10-s − 0.301·11-s − 0.0489·12-s + 1.25·13-s + 1.74·14-s − 0.243·15-s − 0.908·16-s − 1.64·17-s − 0.318·18-s + 0.229·19-s + 0.0357·20-s − 1.05·21-s + 0.288·22-s − 1.70·23-s + 0.599·24-s − 0.821·25-s − 1.19·26-s + 0.192·27-s + 0.154·28-s + ⋯

Functional equation

Λ(s)=(627s/2 ΓC(s) L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 627 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(627s/2 ΓC(s+3/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 627 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 627627    =    3⋅11⋅193 \cdot 11 \cdot 19
Sign: 11
Analytic conductor: 36.994136.9941
Root analytic conductor: 6.082286.08228
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 627, ( :3/2), 1)(2,\ 627,\ (\ :3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 0.54571595420.5457159542
L(12)L(\frac12) ≈\approx 0.54571595420.5457159542
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
11 1+11T 1 + 11T
19 1−19T 1 - 19T
good2 1+2.70T+8T2 1 + 2.70T + 8T^{2}
5 1+4.71T+125T2 1 + 4.71T + 125T^{2}
7 1+33.7T+343T2 1 + 33.7T + 343T^{2}
13 1−58.7T+2.19e3T2 1 - 58.7T + 2.19e3T^{2}
17 1+115.T+4.91e3T2 1 + 115.T + 4.91e3T^{2}
23 1+188.T+1.21e4T2 1 + 188.T + 1.21e4T^{2}
29 1−180.T+2.43e4T2 1 - 180.T + 2.43e4T^{2}
31 1+129.T+2.97e4T2 1 + 129.T + 2.97e4T^{2}
37 1+132.T+5.06e4T2 1 + 132.T + 5.06e4T^{2}
41 1+283.T+6.89e4T2 1 + 283.T + 6.89e4T^{2}
43 1−232.T+7.95e4T2 1 - 232.T + 7.95e4T^{2}
47 1−488.T+1.03e5T2 1 - 488.T + 1.03e5T^{2}
53 1−555.T+1.48e5T2 1 - 555.T + 1.48e5T^{2}
59 1+132.T+2.05e5T2 1 + 132.T + 2.05e5T^{2}
61 1+306.T+2.26e5T2 1 + 306.T + 2.26e5T^{2}
67 1−89.4T+3.00e5T2 1 - 89.4T + 3.00e5T^{2}
71 1−453.T+3.57e5T2 1 - 453.T + 3.57e5T^{2}
73 1+613.T+3.89e5T2 1 + 613.T + 3.89e5T^{2}
79 1+979.T+4.93e5T2 1 + 979.T + 4.93e5T^{2}
83 1−1.33e3T+5.71e5T2 1 - 1.33e3T + 5.71e5T^{2}
89 1−803.T+7.04e5T2 1 - 803.T + 7.04e5T^{2}
97 1−1.61e3T+9.12e5T2 1 - 1.61e3T + 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

−10.13672143801952350953938636450, −9.090648274215548494648120916157, −8.759495386985136272781989942317, −7.77232297747280927456993880909, −6.84563147432029822482199868043, −5.96996627041269426703799078348, −4.22814520671627893702856436251, −3.55726652961332625977487069887, −2.16220435053137599787213651457, −0.47162075064710513122437914262, 0.47162075064710513122437914262, 2.16220435053137599787213651457, 3.55726652961332625977487069887, 4.22814520671627893702856436251, 5.96996627041269426703799078348, 6.84563147432029822482199868043, 7.77232297747280927456993880909, 8.759495386985136272781989942317, 9.090648274215548494648120916157, 10.13672143801952350953938636450

Graph of the ZZ-function along the critical line