↑

Properties

Label 2-693-1.1-c1-0-9
Degree 22
Conductor 693693
Sign −1-1
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 11

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 1.46·2-s + 0.139·4-s − 2.39·5-s − 7-s + 2.72·8-s + 3.50·10-s + 11-s + 5.04·13-s + 1.46·14-s − 4.25·16-s + 6.36·17-s − 5.32·19-s − 0.333·20-s − 1.46·22-s − 4.92·23-s + 0.751·25-s − 7.37·26-s − 0.139·28-s − 5.04·29-s − 7.57·31-s + 0.786·32-s − 9.31·34-s + 2.39·35-s + 4.24·37-s + 7.78·38-s − 6.52·40-s + 0.646·41-s + ⋯
L(s)  = 1  − 1.03·2-s + 0.0695·4-s − 1.07·5-s − 0.377·7-s + 0.962·8-s + 1.10·10-s + 0.301·11-s + 1.39·13-s + 0.390·14-s − 1.06·16-s + 1.54·17-s − 1.22·19-s − 0.0746·20-s − 0.311·22-s − 1.02·23-s + 0.150·25-s − 1.44·26-s − 0.0263·28-s − 0.936·29-s − 1.35·31-s + 0.138·32-s − 1.59·34-s + 0.405·35-s + 0.698·37-s + 1.26·38-s − 1.03·40-s + 0.101·41-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: −1-1
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: 11
Selberg data: (2, 693, ( :1/2), −1)(2,\ 693,\ (\ :1/2),\ -1)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
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+1.46T+2T2 1 + 1.46T + 2T^{2}
5 1+2.39T+5T2 1 + 2.39T + 5T^{2}
13 1−5.04T+13T2 1 - 5.04T + 13T^{2}
17 1−6.36T+17T2 1 - 6.36T + 17T^{2}
19 1+5.32T+19T2 1 + 5.32T + 19T^{2}
23 1+4.92T+23T2 1 + 4.92T + 23T^{2}
29 1+5.04T+29T2 1 + 5.04T + 29T^{2}
31 1+7.57T+31T2 1 + 7.57T + 31T^{2}
37 1−4.24T+37T2 1 - 4.24T + 37T^{2}
41 1−0.646T+41T2 1 - 0.646T + 41T^{2}
43 1+10.5T+43T2 1 + 10.5T + 43T^{2}
47 1+0.526T+47T2 1 + 0.526T + 47T^{2}
53 1+3.72T+53T2 1 + 3.72T + 53T^{2}
59 1+7.97T+59T2 1 + 7.97T + 59T^{2}
61 1+2T+61T2 1 + 2T + 61T^{2}
67 1−8.76T+67T2 1 - 8.76T + 67T^{2}
71 1−11.4T+71T2 1 - 11.4T + 71T^{2}
73 1+13.0T+73T2 1 + 13.0T + 73T^{2}
79 1−11.4T+79T2 1 - 11.4T + 79T^{2}
83 1+13.1T+83T2 1 + 13.1T + 83T^{2}
89 1+11.8T+89T2 1 + 11.8T + 89T^{2}
97 1+1.87T+97T2 1 + 1.87T + 97T^{2}
show more
show less
   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

−9.914766859835927111181720551451, −9.126346120340607902841647364160, −8.190983578807955125415868977757, −7.85545247363002868408878656037, −6.74495981272678135025303279518, −5.64636720839533827401237143803, −4.14500488828332705287493996109, −3.56271916611484593298886659636, −1.54328246457816374798399070843, 0, 1.54328246457816374798399070843, 3.56271916611484593298886659636, 4.14500488828332705287493996109, 5.64636720839533827401237143803, 6.74495981272678135025303279518, 7.85545247363002868408878656037, 8.190983578807955125415868977757, 9.126346120340607902841647364160, 9.914766859835927111181720551451

Graph of the ZZ-function along the critical line