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Properties

Label 2-693-1.1-c3-0-24
Degree 22
Conductor 693693
Sign 11
Analytic cond. 40.888340.8883
Root an. cond. 6.394396.39439
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  − 0.183·2-s − 7.96·4-s + 17.9·5-s − 7·7-s + 2.92·8-s − 3.28·10-s + 11·11-s − 53.2·13-s + 1.28·14-s + 63.1·16-s + 86.7·17-s + 55.6·19-s − 142.·20-s − 2.01·22-s − 185.·23-s + 196.·25-s + 9.75·26-s + 55.7·28-s + 145.·29-s + 132.·31-s − 35.0·32-s − 15.9·34-s − 125.·35-s − 129.·37-s − 10.2·38-s + 52.4·40-s − 139.·41-s + ⋯
L(s)  = 1  − 0.0648·2-s − 0.995·4-s + 1.60·5-s − 0.377·7-s + 0.129·8-s − 0.103·10-s + 0.301·11-s − 1.13·13-s + 0.0245·14-s + 0.987·16-s + 1.23·17-s + 0.672·19-s − 1.59·20-s − 0.0195·22-s − 1.67·23-s + 1.57·25-s + 0.0736·26-s + 0.376·28-s + 0.934·29-s + 0.768·31-s − 0.193·32-s − 0.0802·34-s − 0.606·35-s − 0.574·37-s − 0.0435·38-s + 0.207·40-s − 0.532·41-s + ⋯

Functional equation

Λ(s)=(693s/2 ΓC(s) L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(693s/2 ΓC(s+3/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 693693    =    32⋅7⋅113^{2} \cdot 7 \cdot 11
Sign: 11
Analytic conductor: 40.888340.8883
Root analytic conductor: 6.394396.39439
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 693, ( :3/2), 1)(2,\ 693,\ (\ :3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 1.9660977131.966097713
L(12)L(\frac12) ≈\approx 1.9660977131.966097713
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 1
7 1+7T 1 + 7T
11 1−11T 1 - 11T
good2 1+0.183T+8T2 1 + 0.183T + 8T^{2}
5 1−17.9T+125T2 1 - 17.9T + 125T^{2}
13 1+53.2T+2.19e3T2 1 + 53.2T + 2.19e3T^{2}
17 1−86.7T+4.91e3T2 1 - 86.7T + 4.91e3T^{2}
19 1−55.6T+6.85e3T2 1 - 55.6T + 6.85e3T^{2}
23 1+185.T+1.21e4T2 1 + 185.T + 1.21e4T^{2}
29 1−145.T+2.43e4T2 1 - 145.T + 2.43e4T^{2}
31 1−132.T+2.97e4T2 1 - 132.T + 2.97e4T^{2}
37 1+129.T+5.06e4T2 1 + 129.T + 5.06e4T^{2}
41 1+139.T+6.89e4T2 1 + 139.T + 6.89e4T^{2}
43 1+87.1T+7.95e4T2 1 + 87.1T + 7.95e4T^{2}
47 1−576.T+1.03e5T2 1 - 576.T + 1.03e5T^{2}
53 1−92.0T+1.48e5T2 1 - 92.0T + 1.48e5T^{2}
59 1+315.T+2.05e5T2 1 + 315.T + 2.05e5T^{2}
61 1+146.T+2.26e5T2 1 + 146.T + 2.26e5T^{2}
67 1−734.T+3.00e5T2 1 - 734.T + 3.00e5T^{2}
71 1−702.T+3.57e5T2 1 - 702.T + 3.57e5T^{2}
73 1+361.T+3.89e5T2 1 + 361.T + 3.89e5T^{2}
79 1−1.16e3T+4.93e5T2 1 - 1.16e3T + 4.93e5T^{2}
83 1−1.30e3T+5.71e5T2 1 - 1.30e3T + 5.71e5T^{2}
89 1+1.38e3T+7.04e5T2 1 + 1.38e3T + 7.04e5T^{2}
97 1−212.T+9.12e5T2 1 - 212.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

−9.894647240996706169396942381018, −9.522439691522870505063731410454, −8.519780591081875397332684768419, −7.50586551991173587176439995998, −6.28265185878199681384401012433, −5.54511974197424530591092054552, −4.76699656143625685901496365558, −3.42680860901895258361254926813, −2.16833422925059465524448182676, −0.843632661861296422724326384891, 0.843632661861296422724326384891, 2.16833422925059465524448182676, 3.42680860901895258361254926813, 4.76699656143625685901496365558, 5.54511974197424530591092054552, 6.28265185878199681384401012433, 7.50586551991173587176439995998, 8.519780591081875397332684768419, 9.522439691522870505063731410454, 9.894647240996706169396942381018

Graph of the ZZ-function along the critical line