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Properties

Label 2-693-1.1-c3-0-32
Degree 22
Conductor 693693
Sign −1-1
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 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4.75·2-s + 14.6·4-s − 20.4·5-s + 7·7-s − 31.5·8-s + 97.1·10-s − 11·11-s − 9.41·13-s − 33.2·14-s + 32.8·16-s − 10.5·17-s − 3.48·19-s − 298.·20-s + 52.3·22-s + 50.3·23-s + 292.·25-s + 44.7·26-s + 102.·28-s − 11.5·29-s + 169.·31-s + 95.6·32-s + 50.1·34-s − 142.·35-s − 283.·37-s + 16.5·38-s + 643.·40-s + 165.·41-s + ⋯
L(s)  = 1  − 1.68·2-s + 1.82·4-s − 1.82·5-s + 0.377·7-s − 1.39·8-s + 3.07·10-s − 0.301·11-s − 0.200·13-s − 0.635·14-s + 0.513·16-s − 0.150·17-s − 0.0420·19-s − 3.33·20-s + 0.507·22-s + 0.456·23-s + 2.33·25-s + 0.337·26-s + 0.690·28-s − 0.0740·29-s + 0.980·31-s + 0.528·32-s + 0.252·34-s − 0.690·35-s − 1.26·37-s + 0.0707·38-s + 2.54·40-s + 0.632·41-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: −1-1
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: 11
Selberg data: (2, 693, ( :3/2), −1)(2,\ 693,\ (\ :3/2),\ -1)

Particular Values

L(2)L(2) == 00
L(12)L(\frac12) == 00
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+4.75T+8T2 1 + 4.75T + 8T^{2}
5 1+20.4T+125T2 1 + 20.4T + 125T^{2}
13 1+9.41T+2.19e3T2 1 + 9.41T + 2.19e3T^{2}
17 1+10.5T+4.91e3T2 1 + 10.5T + 4.91e3T^{2}
19 1+3.48T+6.85e3T2 1 + 3.48T + 6.85e3T^{2}
23 1−50.3T+1.21e4T2 1 - 50.3T + 1.21e4T^{2}
29 1+11.5T+2.43e4T2 1 + 11.5T + 2.43e4T^{2}
31 1−169.T+2.97e4T2 1 - 169.T + 2.97e4T^{2}
37 1+283.T+5.06e4T2 1 + 283.T + 5.06e4T^{2}
41 1−165.T+6.89e4T2 1 - 165.T + 6.89e4T^{2}
43 1+209.T+7.95e4T2 1 + 209.T + 7.95e4T^{2}
47 1−604.T+1.03e5T2 1 - 604.T + 1.03e5T^{2}
53 1−164.T+1.48e5T2 1 - 164.T + 1.48e5T^{2}
59 1+292.T+2.05e5T2 1 + 292.T + 2.05e5T^{2}
61 1+202.T+2.26e5T2 1 + 202.T + 2.26e5T^{2}
67 1−750.T+3.00e5T2 1 - 750.T + 3.00e5T^{2}
71 1−14.6T+3.57e5T2 1 - 14.6T + 3.57e5T^{2}
73 1−992.T+3.89e5T2 1 - 992.T + 3.89e5T^{2}
79 1−1.28e3T+4.93e5T2 1 - 1.28e3T + 4.93e5T^{2}
83 1+722.T+5.71e5T2 1 + 722.T + 5.71e5T^{2}
89 1−502.T+7.04e5T2 1 - 502.T + 7.04e5T^{2}
97 1+532.T+9.12e5T2 1 + 532.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

−9.463211053748336953497037445927, −8.571295375157695976911780097941, −8.062202053628427376257503165825, −7.38590706211151393438941735572, −6.68904362127853600676274072233, −4.98854542746674434046658137564, −3.86247563273329270094931333360, −2.57021540552829493167698789347, −1.00093846980919717991116973924, 0, 1.00093846980919717991116973924, 2.57021540552829493167698789347, 3.86247563273329270094931333360, 4.98854542746674434046658137564, 6.68904362127853600676274072233, 7.38590706211151393438941735572, 8.062202053628427376257503165825, 8.571295375157695976911780097941, 9.463211053748336953497037445927

Graph of the ZZ-function along the critical line