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Properties

Label 2-693-1.1-c3-0-46
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  + 5.31·2-s + 20.2·4-s − 7.25·5-s + 7·7-s + 64.8·8-s − 38.5·10-s + 11·11-s + 47.6·13-s + 37.1·14-s + 182.·16-s − 31.5·17-s + 18.9·19-s − 146.·20-s + 58.4·22-s + 200.·23-s − 72.4·25-s + 252.·26-s + 141.·28-s + 224.·29-s − 237.·31-s + 451.·32-s − 167.·34-s − 50.7·35-s + 226.·37-s + 100.·38-s − 470.·40-s − 31.1·41-s + ⋯
L(s)  = 1  + 1.87·2-s + 2.52·4-s − 0.648·5-s + 0.377·7-s + 2.86·8-s − 1.21·10-s + 0.301·11-s + 1.01·13-s + 0.709·14-s + 2.85·16-s − 0.450·17-s + 0.228·19-s − 1.63·20-s + 0.566·22-s + 1.81·23-s − 0.579·25-s + 1.90·26-s + 0.954·28-s + 1.43·29-s − 1.37·31-s + 2.49·32-s − 0.846·34-s − 0.245·35-s + 1.00·37-s + 0.429·38-s − 1.85·40-s − 0.118·41-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 7.0321924817.032192481
L(12)L(\frac12) ≈\approx 7.0321924817.032192481
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 1−5.31T+8T2 1 - 5.31T + 8T^{2}
5 1+7.25T+125T2 1 + 7.25T + 125T^{2}
13 1−47.6T+2.19e3T2 1 - 47.6T + 2.19e3T^{2}
17 1+31.5T+4.91e3T2 1 + 31.5T + 4.91e3T^{2}
19 1−18.9T+6.85e3T2 1 - 18.9T + 6.85e3T^{2}
23 1−200.T+1.21e4T2 1 - 200.T + 1.21e4T^{2}
29 1−224.T+2.43e4T2 1 - 224.T + 2.43e4T^{2}
31 1+237.T+2.97e4T2 1 + 237.T + 2.97e4T^{2}
37 1−226.T+5.06e4T2 1 - 226.T + 5.06e4T^{2}
41 1+31.1T+6.89e4T2 1 + 31.1T + 6.89e4T^{2}
43 1+176.T+7.95e4T2 1 + 176.T + 7.95e4T^{2}
47 1−526.T+1.03e5T2 1 - 526.T + 1.03e5T^{2}
53 1+342.T+1.48e5T2 1 + 342.T + 1.48e5T^{2}
59 1+283.T+2.05e5T2 1 + 283.T + 2.05e5T^{2}
61 1+216.T+2.26e5T2 1 + 216.T + 2.26e5T^{2}
67 1+180.T+3.00e5T2 1 + 180.T + 3.00e5T^{2}
71 1+166.T+3.57e5T2 1 + 166.T + 3.57e5T^{2}
73 1−44.8T+3.89e5T2 1 - 44.8T + 3.89e5T^{2}
79 1−349.T+4.93e5T2 1 - 349.T + 4.93e5T^{2}
83 1−722.T+5.71e5T2 1 - 722.T + 5.71e5T^{2}
89 1−443.T+7.04e5T2 1 - 443.T + 7.04e5T^{2}
97 1+1.80e3T+9.12e5T2 1 + 1.80e3T + 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

−10.75974907252442850699060157896, −9.132205180258756425215627362135, −8.000249797015241956799837335117, −7.09396940228255490626971622069, −6.31855956287401384017756014371, −5.34677099464930394279813634974, −4.46520337912063216056486937970, −3.70873069601795651740437440445, −2.76339494350361259916954255716, −1.33425038097813736915696245208, 1.33425038097813736915696245208, 2.76339494350361259916954255716, 3.70873069601795651740437440445, 4.46520337912063216056486937970, 5.34677099464930394279813634974, 6.31855956287401384017756014371, 7.09396940228255490626971622069, 8.000249797015241956799837335117, 9.132205180258756425215627362135, 10.75974907252442850699060157896

Graph of the ZZ-function along the critical line