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Properties

Label 2-693-1.1-c1-0-13
Degree 22
Conductor 693693
Sign 11
Analytic cond. 5.533635.53363
Root an. cond. 2.352362.35236
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.30·2-s + 3.30·4-s + 5-s − 7-s + 3.00·8-s + 2.30·10-s + 11-s + 3.60·13-s − 2.30·14-s + 0.302·16-s + 4·17-s + 3·19-s + 3.30·20-s + 2.30·22-s + 2·23-s − 4·25-s + 8.30·26-s − 3.30·28-s − 5.60·29-s − 2·31-s − 5.30·32-s + 9.21·34-s − 35-s − 8.21·37-s + 6.90·38-s + 3.00·40-s − 7.21·41-s + ⋯
L(s)  = 1  + 1.62·2-s + 1.65·4-s + 0.447·5-s − 0.377·7-s + 1.06·8-s + 0.728·10-s + 0.301·11-s + 1.00·13-s − 0.615·14-s + 0.0756·16-s + 0.970·17-s + 0.688·19-s + 0.738·20-s + 0.490·22-s + 0.417·23-s − 0.800·25-s + 1.62·26-s − 0.624·28-s − 1.04·29-s − 0.359·31-s − 0.937·32-s + 1.57·34-s − 0.169·35-s − 1.34·37-s + 1.12·38-s + 0.474·40-s − 1.12·41-s + ⋯

Functional equation

Λ(s)=(693s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(693s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s+1/2) \, 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: 5.533635.53363
Root analytic conductor: 2.352362.35236
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 693, ( :1/2), 1)(2,\ 693,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 3.9037499893.903749989
L(12)L(\frac12) ≈\approx 3.9037499893.903749989
L(32)L(\frac{3}{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+T 1 + T
11 1−T 1 - T
good2 1−2.30T+2T2 1 - 2.30T + 2T^{2}
5 1−T+5T2 1 - T + 5T^{2}
13 1−3.60T+13T2 1 - 3.60T + 13T^{2}
17 1−4T+17T2 1 - 4T + 17T^{2}
19 1−3T+19T2 1 - 3T + 19T^{2}
23 1−2T+23T2 1 - 2T + 23T^{2}
29 1+5.60T+29T2 1 + 5.60T + 29T^{2}
31 1+2T+31T2 1 + 2T + 31T^{2}
37 1+8.21T+37T2 1 + 8.21T + 37T^{2}
41 1+7.21T+41T2 1 + 7.21T + 41T^{2}
43 1+5.21T+43T2 1 + 5.21T + 43T^{2}
47 1−2.39T+47T2 1 - 2.39T + 47T^{2}
53 1+53T2 1 + 53T^{2}
59 1−7.60T+59T2 1 - 7.60T + 59T^{2}
61 1+11.2T+61T2 1 + 11.2T + 61T^{2}
67 1−1.60T+67T2 1 - 1.60T + 67T^{2}
71 1−11.2T+71T2 1 - 11.2T + 71T^{2}
73 1+12.8T+73T2 1 + 12.8T + 73T^{2}
79 1+3.21T+79T2 1 + 3.21T + 79T^{2}
83 1−12T+83T2 1 - 12T + 83T^{2}
89 1−6T+89T2 1 - 6T + 89T^{2}
97 1−1.21T+97T2 1 - 1.21T + 97T^{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

−10.70796566540872969329755229258, −9.735986155121920974463703062957, −8.789195515370035215501904505887, −7.47994158733684855720487267118, −6.55917500509980125677895294161, −5.75796334656372880573643349782, −5.13572645756635745181213101822, −3.78340489930993696855598114187, −3.24905057990207045718136250013, −1.75222576013048732290404434874, 1.75222576013048732290404434874, 3.24905057990207045718136250013, 3.78340489930993696855598114187, 5.13572645756635745181213101822, 5.75796334656372880573643349782, 6.55917500509980125677895294161, 7.47994158733684855720487267118, 8.789195515370035215501904505887, 9.735986155121920974463703062957, 10.70796566540872969329755229258

Graph of the ZZ-function along the critical line