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Properties

Label 2-693-1.1-c3-0-1
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  − 5.02·2-s + 17.2·4-s − 20.4·5-s − 7·7-s − 46.2·8-s + 102.·10-s + 11·11-s − 0.0115·13-s + 35.1·14-s + 94.7·16-s + 9.52·17-s − 93.4·19-s − 351.·20-s − 55.2·22-s − 99.9·23-s + 292.·25-s + 0.0580·26-s − 120.·28-s − 276.·29-s − 181.·31-s − 105.·32-s − 47.8·34-s + 143.·35-s − 404.·37-s + 469.·38-s + 946.·40-s + 27.8·41-s + ⋯
L(s)  = 1  − 1.77·2-s + 2.15·4-s − 1.82·5-s − 0.377·7-s − 2.04·8-s + 3.24·10-s + 0.301·11-s − 0.000246·13-s + 0.671·14-s + 1.47·16-s + 0.135·17-s − 1.12·19-s − 3.93·20-s − 0.535·22-s − 0.906·23-s + 2.34·25-s + 0.000437·26-s − 0.813·28-s − 1.76·29-s − 1.05·31-s − 0.581·32-s − 0.241·34-s + 0.690·35-s − 1.79·37-s + 2.00·38-s + 3.73·40-s + 0.105·41-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 0.11365455270.1136545527
L(12)L(\frac12) ≈\approx 0.11365455270.1136545527
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+5.02T+8T2 1 + 5.02T + 8T^{2}
5 1+20.4T+125T2 1 + 20.4T + 125T^{2}
13 1+0.0115T+2.19e3T2 1 + 0.0115T + 2.19e3T^{2}
17 1−9.52T+4.91e3T2 1 - 9.52T + 4.91e3T^{2}
19 1+93.4T+6.85e3T2 1 + 93.4T + 6.85e3T^{2}
23 1+99.9T+1.21e4T2 1 + 99.9T + 1.21e4T^{2}
29 1+276.T+2.43e4T2 1 + 276.T + 2.43e4T^{2}
31 1+181.T+2.97e4T2 1 + 181.T + 2.97e4T^{2}
37 1+404.T+5.06e4T2 1 + 404.T + 5.06e4T^{2}
41 1−27.8T+6.89e4T2 1 - 27.8T + 6.89e4T^{2}
43 1−76.9T+7.95e4T2 1 - 76.9T + 7.95e4T^{2}
47 1−136.T+1.03e5T2 1 - 136.T + 1.03e5T^{2}
53 1+170.T+1.48e5T2 1 + 170.T + 1.48e5T^{2}
59 1−585.T+2.05e5T2 1 - 585.T + 2.05e5T^{2}
61 1+530.T+2.26e5T2 1 + 530.T + 2.26e5T^{2}
67 1+354.T+3.00e5T2 1 + 354.T + 3.00e5T^{2}
71 1+1.11e3T+3.57e5T2 1 + 1.11e3T + 3.57e5T^{2}
73 1+785.T+3.89e5T2 1 + 785.T + 3.89e5T^{2}
79 1+937.T+4.93e5T2 1 + 937.T + 4.93e5T^{2}
83 1+471.T+5.71e5T2 1 + 471.T + 5.71e5T^{2}
89 1+563.T+7.04e5T2 1 + 563.T + 7.04e5T^{2}
97 1−895.T+9.12e5T2 1 - 895.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

−10.04426306200261277331651621119, −8.857511806299228346777237348319, −8.579255538198677495283379369288, −7.39116727870224552855130335791, −7.28848752239255272817711067074, −6.00394131974767877998440514487, −4.23839169270983097455629963554, −3.30401790577683581397018874534, −1.79039089111195507522466622202, −0.24614945773533279077487588749, 0.24614945773533279077487588749, 1.79039089111195507522466622202, 3.30401790577683581397018874534, 4.23839169270983097455629963554, 6.00394131974767877998440514487, 7.28848752239255272817711067074, 7.39116727870224552855130335791, 8.579255538198677495283379369288, 8.857511806299228346777237348319, 10.04426306200261277331651621119

Graph of the ZZ-function along the critical line