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Properties

Label 2-693-1.1-c3-0-28
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  − 1.59·2-s − 5.44·4-s + 17.2·5-s + 7·7-s + 21.5·8-s − 27.6·10-s − 11·11-s + 46.1·13-s − 11.1·14-s + 9.11·16-s − 19.8·17-s + 76.5·19-s − 93.9·20-s + 17.5·22-s + 163.·23-s + 173.·25-s − 73.7·26-s − 38.0·28-s − 158.·29-s + 170.·31-s − 186.·32-s + 31.7·34-s + 120.·35-s − 245.·37-s − 122.·38-s + 371.·40-s + 3.33·41-s + ⋯
L(s)  = 1  − 0.565·2-s − 0.680·4-s + 1.54·5-s + 0.377·7-s + 0.950·8-s − 0.874·10-s − 0.301·11-s + 0.983·13-s − 0.213·14-s + 0.142·16-s − 0.283·17-s + 0.924·19-s − 1.05·20-s + 0.170·22-s + 1.47·23-s + 1.38·25-s − 0.556·26-s − 0.257·28-s − 1.01·29-s + 0.989·31-s − 1.03·32-s + 0.160·34-s + 0.584·35-s − 1.09·37-s − 0.522·38-s + 1.46·40-s + 0.0127·41-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 1.9418504511.941850451
L(12)L(\frac12) ≈\approx 1.9418504511.941850451
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+1.59T+8T2 1 + 1.59T + 8T^{2}
5 1−17.2T+125T2 1 - 17.2T + 125T^{2}
13 1−46.1T+2.19e3T2 1 - 46.1T + 2.19e3T^{2}
17 1+19.8T+4.91e3T2 1 + 19.8T + 4.91e3T^{2}
19 1−76.5T+6.85e3T2 1 - 76.5T + 6.85e3T^{2}
23 1−163.T+1.21e4T2 1 - 163.T + 1.21e4T^{2}
29 1+158.T+2.43e4T2 1 + 158.T + 2.43e4T^{2}
31 1−170.T+2.97e4T2 1 - 170.T + 2.97e4T^{2}
37 1+245.T+5.06e4T2 1 + 245.T + 5.06e4T^{2}
41 1−3.33T+6.89e4T2 1 - 3.33T + 6.89e4T^{2}
43 1+122.T+7.95e4T2 1 + 122.T + 7.95e4T^{2}
47 1+390.T+1.03e5T2 1 + 390.T + 1.03e5T^{2}
53 1−410.T+1.48e5T2 1 - 410.T + 1.48e5T^{2}
59 1+408.T+2.05e5T2 1 + 408.T + 2.05e5T^{2}
61 1+21.9T+2.26e5T2 1 + 21.9T + 2.26e5T^{2}
67 1−618.T+3.00e5T2 1 - 618.T + 3.00e5T^{2}
71 1+929.T+3.57e5T2 1 + 929.T + 3.57e5T^{2}
73 1−868.T+3.89e5T2 1 - 868.T + 3.89e5T^{2}
79 1−152.T+4.93e5T2 1 - 152.T + 4.93e5T^{2}
83 1−100.T+5.71e5T2 1 - 100.T + 5.71e5T^{2}
89 1−1.06e3T+7.04e5T2 1 - 1.06e3T + 7.04e5T^{2}
97 1−1.41e3T+9.12e5T2 1 - 1.41e3T + 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

−9.975134798553365431949019210853, −9.153512138391453150467368880748, −8.669914983972517088752935633477, −7.59528417470824914659072702850, −6.49827509700104427821851259377, −5.45156880061480114105859377419, −4.84988767304440927799871452842, −3.35576340975389550884811278914, −1.88441430002707892073965282103, −0.954094278141210649750337880694, 0.954094278141210649750337880694, 1.88441430002707892073965282103, 3.35576340975389550884811278914, 4.84988767304440927799871452842, 5.45156880061480114105859377419, 6.49827509700104427821851259377, 7.59528417470824914659072702850, 8.669914983972517088752935633477, 9.153512138391453150467368880748, 9.975134798553365431949019210853

Graph of the ZZ-function along the critical line