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Properties

Label 2-693-1.1-c3-0-22
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  + 3.63·2-s + 5.18·4-s − 21.1·5-s + 7·7-s − 10.2·8-s − 76.6·10-s − 11·11-s + 87.3·13-s + 25.4·14-s − 78.6·16-s + 28.2·17-s − 97.9·19-s − 109.·20-s − 39.9·22-s + 112.·23-s + 320.·25-s + 317.·26-s + 36.2·28-s − 14.9·29-s + 138.·31-s − 203.·32-s + 102.·34-s − 147.·35-s + 206.·37-s − 355.·38-s + 215.·40-s − 321.·41-s + ⋯
L(s)  = 1  + 1.28·2-s + 0.647·4-s − 1.88·5-s + 0.377·7-s − 0.452·8-s − 2.42·10-s − 0.301·11-s + 1.86·13-s + 0.485·14-s − 1.22·16-s + 0.403·17-s − 1.18·19-s − 1.22·20-s − 0.387·22-s + 1.01·23-s + 2.56·25-s + 2.39·26-s + 0.244·28-s − 0.0955·29-s + 0.802·31-s − 1.12·32-s + 0.517·34-s − 0.713·35-s + 0.919·37-s − 1.51·38-s + 0.853·40-s − 1.22·41-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 2.6223703172.622370317
L(12)L(\frac12) ≈\approx 2.6223703172.622370317
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 1−3.63T+8T2 1 - 3.63T + 8T^{2}
5 1+21.1T+125T2 1 + 21.1T + 125T^{2}
13 1−87.3T+2.19e3T2 1 - 87.3T + 2.19e3T^{2}
17 1−28.2T+4.91e3T2 1 - 28.2T + 4.91e3T^{2}
19 1+97.9T+6.85e3T2 1 + 97.9T + 6.85e3T^{2}
23 1−112.T+1.21e4T2 1 - 112.T + 1.21e4T^{2}
29 1+14.9T+2.43e4T2 1 + 14.9T + 2.43e4T^{2}
31 1−138.T+2.97e4T2 1 - 138.T + 2.97e4T^{2}
37 1−206.T+5.06e4T2 1 - 206.T + 5.06e4T^{2}
41 1+321.T+6.89e4T2 1 + 321.T + 6.89e4T^{2}
43 1−285.T+7.95e4T2 1 - 285.T + 7.95e4T^{2}
47 1−303.T+1.03e5T2 1 - 303.T + 1.03e5T^{2}
53 1−554.T+1.48e5T2 1 - 554.T + 1.48e5T^{2}
59 1−693.T+2.05e5T2 1 - 693.T + 2.05e5T^{2}
61 1−156.T+2.26e5T2 1 - 156.T + 2.26e5T^{2}
67 1−584.T+3.00e5T2 1 - 584.T + 3.00e5T^{2}
71 1+363.T+3.57e5T2 1 + 363.T + 3.57e5T^{2}
73 1+747.T+3.89e5T2 1 + 747.T + 3.89e5T^{2}
79 1−419.T+4.93e5T2 1 - 419.T + 4.93e5T^{2}
83 1+1.17e3T+5.71e5T2 1 + 1.17e3T + 5.71e5T^{2}
89 1−397.T+7.04e5T2 1 - 397.T + 7.04e5T^{2}
97 1+1.33e3T+9.12e5T2 1 + 1.33e3T + 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

−10.53546410758396223170220680306, −8.703004181212584552021084316538, −8.443741241858736150161418802768, −7.30874993226106632022443840921, −6.38434472266800392644747030944, −5.30991150451472222951706877945, −4.23450970235591637826271717707, −3.86082219982395648308233352338, −2.84676433370662967029939020265, −0.77938499790889467037464318533, 0.77938499790889467037464318533, 2.84676433370662967029939020265, 3.86082219982395648308233352338, 4.23450970235591637826271717707, 5.30991150451472222951706877945, 6.38434472266800392644747030944, 7.30874993226106632022443840921, 8.443741241858736150161418802768, 8.703004181212584552021084316538, 10.53546410758396223170220680306

Graph of the ZZ-function along the critical line