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Properties

Label 2-693-1.1-c1-0-23
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

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.93·2-s + 1.74·4-s − 4.18·5-s − 7-s − 0.491·8-s − 8.10·10-s + 11-s − 3.17·13-s − 1.93·14-s − 4.44·16-s − 6.85·17-s − 0.318·19-s − 7.31·20-s + 1.93·22-s + 1.87·23-s + 12.5·25-s − 6.14·26-s − 1.74·28-s + 3.17·29-s + 9.23·31-s − 7.61·32-s − 13.2·34-s + 4.18·35-s − 7.55·37-s − 0.616·38-s + 2.06·40-s − 9.36·41-s + ⋯
L(s)  = 1  + 1.36·2-s + 0.872·4-s − 1.87·5-s − 0.377·7-s − 0.173·8-s − 2.56·10-s + 0.301·11-s − 0.880·13-s − 0.517·14-s − 1.11·16-s − 1.66·17-s − 0.0731·19-s − 1.63·20-s + 0.412·22-s + 0.390·23-s + 2.51·25-s − 1.20·26-s − 0.329·28-s + 0.589·29-s + 1.65·31-s − 1.34·32-s − 2.27·34-s + 0.708·35-s − 1.24·37-s − 0.100·38-s + 0.325·40-s − 1.46·41-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 1−1.93T+2T2 1 - 1.93T + 2T^{2}
5 1+4.18T+5T2 1 + 4.18T + 5T^{2}
13 1+3.17T+13T2 1 + 3.17T + 13T^{2}
17 1+6.85T+17T2 1 + 6.85T + 17T^{2}
19 1+0.318T+19T2 1 + 0.318T + 19T^{2}
23 1−1.87T+23T2 1 - 1.87T + 23T^{2}
29 1−3.17T+29T2 1 - 3.17T + 29T^{2}
31 1−9.23T+31T2 1 - 9.23T + 31T^{2}
37 1+7.55T+37T2 1 + 7.55T + 37T^{2}
41 1+9.36T+41T2 1 + 9.36T + 41T^{2}
43 1+10.8T+43T2 1 + 10.8T + 43T^{2}
47 1−8.06T+47T2 1 - 8.06T + 47T^{2}
53 1+0.508T+53T2 1 + 0.508T + 53T^{2}
59 1−7.04T+59T2 1 - 7.04T + 59T^{2}
61 1+2T+61T2 1 + 2T + 61T^{2}
67 1+2.66T+67T2 1 + 2.66T + 67T^{2}
71 1−5.01T+71T2 1 - 5.01T + 71T^{2}
73 1+4.82T+73T2 1 + 4.82T + 73T^{2}
79 1−5.01T+79T2 1 - 5.01T + 79T^{2}
83 1+3.52T+83T2 1 + 3.52T + 83T^{2}
89 1−1.74T+89T2 1 - 1.74T + 89T^{2}
97 1+12.2T+97T2 1 + 12.2T + 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

−10.31892881686011013076334032188, −8.936466461282421859104530205200, −8.266687228494963366525791727658, −6.95467925429763141050855852171, −6.65750338714462598945729693110, −5.02634003713131708178759643517, −4.45121417313434353743955773395, −3.63280981579582015582749737956, −2.70785586088951357545083744290, 0, 2.70785586088951357545083744290, 3.63280981579582015582749737956, 4.45121417313434353743955773395, 5.02634003713131708178759643517, 6.65750338714462598945729693110, 6.95467925429763141050855852171, 8.266687228494963366525791727658, 8.936466461282421859104530205200, 10.31892881686011013076334032188

Graph of the ZZ-function along the critical line