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Properties

Label 2-693-1.1-c3-0-12
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  − 1.50·2-s − 5.73·4-s + 0.155·5-s − 7·7-s + 20.6·8-s − 0.233·10-s + 11·11-s + 88.7·13-s + 10.5·14-s + 14.7·16-s − 102.·17-s − 26.8·19-s − 0.890·20-s − 16.5·22-s − 65.1·23-s − 124.·25-s − 133.·26-s + 40.1·28-s − 136.·29-s − 87.8·31-s − 187.·32-s + 153.·34-s − 1.08·35-s + 391.·37-s + 40.4·38-s + 3.20·40-s + 69.8·41-s + ⋯
L(s)  = 1  − 0.532·2-s − 0.716·4-s + 0.0138·5-s − 0.377·7-s + 0.913·8-s − 0.00739·10-s + 0.301·11-s + 1.89·13-s + 0.201·14-s + 0.230·16-s − 1.45·17-s − 0.324·19-s − 0.00995·20-s − 0.160·22-s − 0.590·23-s − 0.999·25-s − 1.00·26-s + 0.270·28-s − 0.874·29-s − 0.508·31-s − 1.03·32-s + 0.775·34-s − 0.00524·35-s + 1.73·37-s + 0.172·38-s + 0.0126·40-s + 0.266·41-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 1.0389973531.038997353
L(12)L(\frac12) ≈\approx 1.0389973531.038997353
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+1.50T+8T2 1 + 1.50T + 8T^{2}
5 1−0.155T+125T2 1 - 0.155T + 125T^{2}
13 1−88.7T+2.19e3T2 1 - 88.7T + 2.19e3T^{2}
17 1+102.T+4.91e3T2 1 + 102.T + 4.91e3T^{2}
19 1+26.8T+6.85e3T2 1 + 26.8T + 6.85e3T^{2}
23 1+65.1T+1.21e4T2 1 + 65.1T + 1.21e4T^{2}
29 1+136.T+2.43e4T2 1 + 136.T + 2.43e4T^{2}
31 1+87.8T+2.97e4T2 1 + 87.8T + 2.97e4T^{2}
37 1−391.T+5.06e4T2 1 - 391.T + 5.06e4T^{2}
41 1−69.8T+6.89e4T2 1 - 69.8T + 6.89e4T^{2}
43 1−293.T+7.95e4T2 1 - 293.T + 7.95e4T^{2}
47 1+122.T+1.03e5T2 1 + 122.T + 1.03e5T^{2}
53 1+140.T+1.48e5T2 1 + 140.T + 1.48e5T^{2}
59 1−653.T+2.05e5T2 1 - 653.T + 2.05e5T^{2}
61 1−295.T+2.26e5T2 1 - 295.T + 2.26e5T^{2}
67 1+82.0T+3.00e5T2 1 + 82.0T + 3.00e5T^{2}
71 1−579.T+3.57e5T2 1 - 579.T + 3.57e5T^{2}
73 1+123.T+3.89e5T2 1 + 123.T + 3.89e5T^{2}
79 1−420.T+4.93e5T2 1 - 420.T + 4.93e5T^{2}
83 1+59.7T+5.71e5T2 1 + 59.7T + 5.71e5T^{2}
89 1−280.T+7.04e5T2 1 - 280.T + 7.04e5T^{2}
97 1−19.1T+9.12e5T2 1 - 19.1T + 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

−9.865219278374624669867951696012, −9.137723879258319793347386106311, −8.532911566815249268145027707022, −7.68655387436397038856427235205, −6.46996501872286748473572509047, −5.72917317210509237914354272474, −4.28388123289865148516018119376, −3.73443347680024143159591878249, −1.97199235079222243830832791133, −0.64850170131321244074605860951, 0.64850170131321244074605860951, 1.97199235079222243830832791133, 3.73443347680024143159591878249, 4.28388123289865148516018119376, 5.72917317210509237914354272474, 6.46996501872286748473572509047, 7.68655387436397038856427235205, 8.532911566815249268145027707022, 9.137723879258319793347386106311, 9.865219278374624669867951696012

Graph of the ZZ-function along the critical line