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Properties

Label 2-693-1.1-c3-0-41
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  + 3.56·2-s + 4.68·4-s + 15.6·5-s + 7·7-s − 11.8·8-s + 55.8·10-s − 11·11-s + 52.9·13-s + 24.9·14-s − 79.5·16-s + 77.1·17-s + 13.4·19-s + 73.4·20-s − 39.1·22-s + 59.0·23-s + 121.·25-s + 188.·26-s + 32.7·28-s + 69.3·29-s + 75.9·31-s − 188.·32-s + 274.·34-s + 109.·35-s − 335.·37-s + 47.7·38-s − 185.·40-s + 318.·41-s + ⋯
L(s)  = 1  + 1.25·2-s + 0.585·4-s + 1.40·5-s + 0.377·7-s − 0.521·8-s + 1.76·10-s − 0.301·11-s + 1.12·13-s + 0.475·14-s − 1.24·16-s + 1.10·17-s + 0.161·19-s + 0.821·20-s − 0.379·22-s + 0.535·23-s + 0.968·25-s + 1.42·26-s + 0.221·28-s + 0.443·29-s + 0.440·31-s − 1.04·32-s + 1.38·34-s + 0.530·35-s − 1.49·37-s + 0.203·38-s − 0.732·40-s + 1.21·41-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 5.3192646125.319264612
L(12)L(\frac12) ≈\approx 5.3192646125.319264612
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 1−3.56T+8T2 1 - 3.56T + 8T^{2}
5 1−15.6T+125T2 1 - 15.6T + 125T^{2}
13 1−52.9T+2.19e3T2 1 - 52.9T + 2.19e3T^{2}
17 1−77.1T+4.91e3T2 1 - 77.1T + 4.91e3T^{2}
19 1−13.4T+6.85e3T2 1 - 13.4T + 6.85e3T^{2}
23 1−59.0T+1.21e4T2 1 - 59.0T + 1.21e4T^{2}
29 1−69.3T+2.43e4T2 1 - 69.3T + 2.43e4T^{2}
31 1−75.9T+2.97e4T2 1 - 75.9T + 2.97e4T^{2}
37 1+335.T+5.06e4T2 1 + 335.T + 5.06e4T^{2}
41 1−318.T+6.89e4T2 1 - 318.T + 6.89e4T^{2}
43 1+57.2T+7.95e4T2 1 + 57.2T + 7.95e4T^{2}
47 1−577.T+1.03e5T2 1 - 577.T + 1.03e5T^{2}
53 1+315.T+1.48e5T2 1 + 315.T + 1.48e5T^{2}
59 1−598.T+2.05e5T2 1 - 598.T + 2.05e5T^{2}
61 1+337.T+2.26e5T2 1 + 337.T + 2.26e5T^{2}
67 1+107.T+3.00e5T2 1 + 107.T + 3.00e5T^{2}
71 1+405.T+3.57e5T2 1 + 405.T + 3.57e5T^{2}
73 1+133.T+3.89e5T2 1 + 133.T + 3.89e5T^{2}
79 1+922.T+4.93e5T2 1 + 922.T + 4.93e5T^{2}
83 1−1.22e3T+5.71e5T2 1 - 1.22e3T + 5.71e5T^{2}
89 1+1.58e3T+7.04e5T2 1 + 1.58e3T + 7.04e5T^{2}
97 1+287.T+9.12e5T2 1 + 287.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

−10.17994027949927581836812395125, −9.238183426258638488630926694384, −8.449384439413613906694662761847, −7.10078077422777306636968571432, −5.98695121282127991147654217037, −5.63209837978939296006271690091, −4.71271466931354847951921708772, −3.52784013374602916837571307139, −2.52300906503682964861478311589, −1.24021236054099616543984647204, 1.24021236054099616543984647204, 2.52300906503682964861478311589, 3.52784013374602916837571307139, 4.71271466931354847951921708772, 5.63209837978939296006271690091, 5.98695121282127991147654217037, 7.10078077422777306636968571432, 8.449384439413613906694662761847, 9.238183426258638488630926694384, 10.17994027949927581836812395125

Graph of the ZZ-function along the critical line