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Properties

Label 2-693-1.1-c3-0-15
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  − 4.54·2-s + 12.6·4-s − 6.53·5-s + 7·7-s − 21.0·8-s + 29.6·10-s − 11·11-s + 71.3·13-s − 31.8·14-s − 5.29·16-s − 2.45·17-s + 80.0·19-s − 82.6·20-s + 49.9·22-s − 61.8·23-s − 82.2·25-s − 324.·26-s + 88.5·28-s + 156.·29-s + 77.7·31-s + 192.·32-s + 11.1·34-s − 45.7·35-s + 84.7·37-s − 363.·38-s + 137.·40-s − 28.8·41-s + ⋯
L(s)  = 1  − 1.60·2-s + 1.58·4-s − 0.584·5-s + 0.377·7-s − 0.932·8-s + 0.939·10-s − 0.301·11-s + 1.52·13-s − 0.607·14-s − 0.0827·16-s − 0.0349·17-s + 0.966·19-s − 0.923·20-s + 0.484·22-s − 0.560·23-s − 0.658·25-s − 2.44·26-s + 0.597·28-s + 1.00·29-s + 0.450·31-s + 1.06·32-s + 0.0561·34-s − 0.220·35-s + 0.376·37-s − 1.55·38-s + 0.545·40-s − 0.109·41-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 0.80192631430.8019263143
L(12)L(\frac12) ≈\approx 0.80192631430.8019263143
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+4.54T+8T2 1 + 4.54T + 8T^{2}
5 1+6.53T+125T2 1 + 6.53T + 125T^{2}
13 1−71.3T+2.19e3T2 1 - 71.3T + 2.19e3T^{2}
17 1+2.45T+4.91e3T2 1 + 2.45T + 4.91e3T^{2}
19 1−80.0T+6.85e3T2 1 - 80.0T + 6.85e3T^{2}
23 1+61.8T+1.21e4T2 1 + 61.8T + 1.21e4T^{2}
29 1−156.T+2.43e4T2 1 - 156.T + 2.43e4T^{2}
31 1−77.7T+2.97e4T2 1 - 77.7T + 2.97e4T^{2}
37 1−84.7T+5.06e4T2 1 - 84.7T + 5.06e4T^{2}
41 1+28.8T+6.89e4T2 1 + 28.8T + 6.89e4T^{2}
43 1+352.T+7.95e4T2 1 + 352.T + 7.95e4T^{2}
47 1−256.T+1.03e5T2 1 - 256.T + 1.03e5T^{2}
53 1+492.T+1.48e5T2 1 + 492.T + 1.48e5T^{2}
59 1−3.12T+2.05e5T2 1 - 3.12T + 2.05e5T^{2}
61 1+159.T+2.26e5T2 1 + 159.T + 2.26e5T^{2}
67 1+521.T+3.00e5T2 1 + 521.T + 3.00e5T^{2}
71 1−885.T+3.57e5T2 1 - 885.T + 3.57e5T^{2}
73 1+375.T+3.89e5T2 1 + 375.T + 3.89e5T^{2}
79 1+1.34e3T+4.93e5T2 1 + 1.34e3T + 4.93e5T^{2}
83 1−1.37e3T+5.71e5T2 1 - 1.37e3T + 5.71e5T^{2}
89 1−111.T+7.04e5T2 1 - 111.T + 7.04e5T^{2}
97 1−472.T+9.12e5T2 1 - 472.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

−10.00245988880718076219434094945, −9.115583821861748867683411508412, −8.221461076506843792233610510284, −7.919897551605345169482123350504, −6.87619158569995011516680490295, −5.89344698451128017408896325060, −4.46713710308949712510447831323, −3.19799877119024357063671321045, −1.72084336848014269129520919586, −0.68478620519020112135929047380, 0.68478620519020112135929047380, 1.72084336848014269129520919586, 3.19799877119024357063671321045, 4.46713710308949712510447831323, 5.89344698451128017408896325060, 6.87619158569995011516680490295, 7.919897551605345169482123350504, 8.221461076506843792233610510284, 9.115583821861748867683411508412, 10.00245988880718076219434094945

Graph of the ZZ-function along the critical line