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Properties

Label 2-693-1.1-c3-0-16
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  − 0.561·2-s − 7.68·4-s + 3.31·5-s + 7·7-s + 8.80·8-s − 1.86·10-s − 11·11-s − 41.9·13-s − 3.93·14-s + 56.5·16-s + 68.8·17-s − 114.·19-s − 25.4·20-s + 6.17·22-s + 124.·23-s − 114.·25-s + 23.5·26-s − 53.7·28-s + 147.·29-s − 55.9·31-s − 102.·32-s − 38.6·34-s + 23.2·35-s + 162.·37-s + 64.2·38-s + 29.2·40-s − 258.·41-s + ⋯
L(s)  = 1  − 0.198·2-s − 0.960·4-s + 0.296·5-s + 0.377·7-s + 0.389·8-s − 0.0588·10-s − 0.301·11-s − 0.894·13-s − 0.0750·14-s + 0.883·16-s + 0.982·17-s − 1.38·19-s − 0.284·20-s + 0.0598·22-s + 1.13·23-s − 0.912·25-s + 0.177·26-s − 0.363·28-s + 0.945·29-s − 0.324·31-s − 0.564·32-s − 0.195·34-s + 0.112·35-s + 0.724·37-s + 0.274·38-s + 0.115·40-s − 0.985·41-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 1.3055983271.305598327
L(12)L(\frac12) ≈\approx 1.3055983271.305598327
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+0.561T+8T2 1 + 0.561T + 8T^{2}
5 1−3.31T+125T2 1 - 3.31T + 125T^{2}
13 1+41.9T+2.19e3T2 1 + 41.9T + 2.19e3T^{2}
17 1−68.8T+4.91e3T2 1 - 68.8T + 4.91e3T^{2}
19 1+114.T+6.85e3T2 1 + 114.T + 6.85e3T^{2}
23 1−124.T+1.21e4T2 1 - 124.T + 1.21e4T^{2}
29 1−147.T+2.43e4T2 1 - 147.T + 2.43e4T^{2}
31 1+55.9T+2.97e4T2 1 + 55.9T + 2.97e4T^{2}
37 1−162.T+5.06e4T2 1 - 162.T + 5.06e4T^{2}
41 1+258.T+6.89e4T2 1 + 258.T + 6.89e4T^{2}
43 1+106.T+7.95e4T2 1 + 106.T + 7.95e4T^{2}
47 1+110.T+1.03e5T2 1 + 110.T + 1.03e5T^{2}
53 1+10.4T+1.48e5T2 1 + 10.4T + 1.48e5T^{2}
59 1−182.T+2.05e5T2 1 - 182.T + 2.05e5T^{2}
61 1−189.T+2.26e5T2 1 - 189.T + 2.26e5T^{2}
67 1−580.T+3.00e5T2 1 - 580.T + 3.00e5T^{2}
71 1−1.16e3T+3.57e5T2 1 - 1.16e3T + 3.57e5T^{2}
73 1+79.6T+3.89e5T2 1 + 79.6T + 3.89e5T^{2}
79 1−1.09e3T+4.93e5T2 1 - 1.09e3T + 4.93e5T^{2}
83 1−874.T+5.71e5T2 1 - 874.T + 5.71e5T^{2}
89 1−844.T+7.04e5T2 1 - 844.T + 7.04e5T^{2}
97 1−925.T+9.12e5T2 1 - 925.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

−10.00193232648248253138585710998, −9.262291741723733578835309238229, −8.329262106372243706091523440451, −7.71149394828181567647543638257, −6.52097645129530690922416049163, −5.30800327038205900887971515050, −4.73033359160660884453408105521, −3.54747174217649852290930377969, −2.13387747067743997429406781782, −0.68492672600460339496107357010, 0.68492672600460339496107357010, 2.13387747067743997429406781782, 3.54747174217649852290930377969, 4.73033359160660884453408105521, 5.30800327038205900887971515050, 6.52097645129530690922416049163, 7.71149394828181567647543638257, 8.329262106372243706091523440451, 9.262291741723733578835309238229, 10.00193232648248253138585710998

Graph of the ZZ-function along the critical line