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Properties

Label 2-693-1.1-c3-0-48
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  + 4.79·2-s + 14.9·4-s + 6.25·5-s + 7·7-s + 33.4·8-s + 29.9·10-s − 11·11-s − 35.7·13-s + 33.5·14-s + 40.4·16-s + 133.·17-s + 161.·19-s + 93.6·20-s − 52.7·22-s + 66.2·23-s − 85.8·25-s − 171.·26-s + 104.·28-s + 208.·29-s − 39.3·31-s − 73.5·32-s + 637.·34-s + 43.7·35-s + 197.·37-s + 774.·38-s + 209.·40-s − 434.·41-s + ⋯
L(s)  = 1  + 1.69·2-s + 1.87·4-s + 0.559·5-s + 0.377·7-s + 1.47·8-s + 0.947·10-s − 0.301·11-s − 0.763·13-s + 0.640·14-s + 0.632·16-s + 1.89·17-s + 1.95·19-s + 1.04·20-s − 0.510·22-s + 0.600·23-s − 0.687·25-s − 1.29·26-s + 0.707·28-s + 1.33·29-s − 0.228·31-s − 0.406·32-s + 3.21·34-s + 0.211·35-s + 0.877·37-s + 3.30·38-s + 0.826·40-s − 1.65·41-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 6.7840385146.784038514
L(12)L(\frac12) ≈\approx 6.7840385146.784038514
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 1−4.79T+8T2 1 - 4.79T + 8T^{2}
5 1−6.25T+125T2 1 - 6.25T + 125T^{2}
13 1+35.7T+2.19e3T2 1 + 35.7T + 2.19e3T^{2}
17 1−133.T+4.91e3T2 1 - 133.T + 4.91e3T^{2}
19 1−161.T+6.85e3T2 1 - 161.T + 6.85e3T^{2}
23 1−66.2T+1.21e4T2 1 - 66.2T + 1.21e4T^{2}
29 1−208.T+2.43e4T2 1 - 208.T + 2.43e4T^{2}
31 1+39.3T+2.97e4T2 1 + 39.3T + 2.97e4T^{2}
37 1−197.T+5.06e4T2 1 - 197.T + 5.06e4T^{2}
41 1+434.T+6.89e4T2 1 + 434.T + 6.89e4T^{2}
43 1−375.T+7.95e4T2 1 - 375.T + 7.95e4T^{2}
47 1+503.T+1.03e5T2 1 + 503.T + 1.03e5T^{2}
53 1+44.8T+1.48e5T2 1 + 44.8T + 1.48e5T^{2}
59 1+582.T+2.05e5T2 1 + 582.T + 2.05e5T^{2}
61 1−73.2T+2.26e5T2 1 - 73.2T + 2.26e5T^{2}
67 1+928.T+3.00e5T2 1 + 928.T + 3.00e5T^{2}
71 1−755.T+3.57e5T2 1 - 755.T + 3.57e5T^{2}
73 1−277.T+3.89e5T2 1 - 277.T + 3.89e5T^{2}
79 1−651.T+4.93e5T2 1 - 651.T + 4.93e5T^{2}
83 1+282.T+5.71e5T2 1 + 282.T + 5.71e5T^{2}
89 1+1.04e3T+7.04e5T2 1 + 1.04e3T + 7.04e5T^{2}
97 1−1.11e3T+9.12e5T2 1 - 1.11e3T + 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

−10.14257513538614044281169621044, −9.509688724506428169826359248300, −7.927201322560204273781636084615, −7.25382942322055545933398159569, −6.10846308542284062292702327809, −5.31179857460304737958220479914, −4.84052083323305641998157362567, −3.44277687262179239301347908663, −2.73220088095683067244704688146, −1.34660957046689371538856528282, 1.34660957046689371538856528282, 2.73220088095683067244704688146, 3.44277687262179239301347908663, 4.84052083323305641998157362567, 5.31179857460304737958220479914, 6.10846308542284062292702327809, 7.25382942322055545933398159569, 7.927201322560204273781636084615, 9.509688724506428169826359248300, 10.14257513538614044281169621044

Graph of the ZZ-function along the critical line