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Properties

Label 2-693-1.1-c3-0-40
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  − 0.221·2-s − 7.95·4-s − 15.1·5-s + 7·7-s + 3.53·8-s + 3.36·10-s − 11·11-s + 37.8·13-s − 1.55·14-s + 62.8·16-s + 61.7·17-s + 54.5·19-s + 120.·20-s + 2.44·22-s − 24.4·23-s + 104.·25-s − 8.38·26-s − 55.6·28-s + 16.2·29-s − 190.·31-s − 42.2·32-s − 13.7·34-s − 106.·35-s + 170.·37-s − 12.1·38-s − 53.6·40-s − 78.0·41-s + ⋯
L(s)  = 1  − 0.0784·2-s − 0.993·4-s − 1.35·5-s + 0.377·7-s + 0.156·8-s + 0.106·10-s − 0.301·11-s + 0.806·13-s − 0.0296·14-s + 0.981·16-s + 0.881·17-s + 0.658·19-s + 1.34·20-s + 0.0236·22-s − 0.222·23-s + 0.838·25-s − 0.0632·26-s − 0.375·28-s + 0.104·29-s − 1.10·31-s − 0.233·32-s − 0.0691·34-s − 0.512·35-s + 0.758·37-s − 0.0516·38-s − 0.212·40-s − 0.297·41-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+0.221T+8T2 1 + 0.221T + 8T^{2}
5 1+15.1T+125T2 1 + 15.1T + 125T^{2}
13 1−37.8T+2.19e3T2 1 - 37.8T + 2.19e3T^{2}
17 1−61.7T+4.91e3T2 1 - 61.7T + 4.91e3T^{2}
19 1−54.5T+6.85e3T2 1 - 54.5T + 6.85e3T^{2}
23 1+24.4T+1.21e4T2 1 + 24.4T + 1.21e4T^{2}
29 1−16.2T+2.43e4T2 1 - 16.2T + 2.43e4T^{2}
31 1+190.T+2.97e4T2 1 + 190.T + 2.97e4T^{2}
37 1−170.T+5.06e4T2 1 - 170.T + 5.06e4T^{2}
41 1+78.0T+6.89e4T2 1 + 78.0T + 6.89e4T^{2}
43 1−45.5T+7.95e4T2 1 - 45.5T + 7.95e4T^{2}
47 1+273.T+1.03e5T2 1 + 273.T + 1.03e5T^{2}
53 1−163.T+1.48e5T2 1 - 163.T + 1.48e5T^{2}
59 1+650.T+2.05e5T2 1 + 650.T + 2.05e5T^{2}
61 1−257.T+2.26e5T2 1 - 257.T + 2.26e5T^{2}
67 1+399.T+3.00e5T2 1 + 399.T + 3.00e5T^{2}
71 1+198.T+3.57e5T2 1 + 198.T + 3.57e5T^{2}
73 1+226.T+3.89e5T2 1 + 226.T + 3.89e5T^{2}
79 1−138.T+4.93e5T2 1 - 138.T + 4.93e5T^{2}
83 1+490.T+5.71e5T2 1 + 490.T + 5.71e5T^{2}
89 1+221.T+7.04e5T2 1 + 221.T + 7.04e5T^{2}
97 1−1.59e3T+9.12e5T2 1 - 1.59e3T + 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

−9.541503303718091858844022581433, −8.621373122572717417447165366548, −7.949288001457336765291419684749, −7.38359355245444053310585557251, −5.86207681602135082004995696436, −4.90853663868747665439950088080, −3.98686009375422435268885149842, −3.26317164093103664574894566988, −1.20295292431872029950449250423, 0, 1.20295292431872029950449250423, 3.26317164093103664574894566988, 3.98686009375422435268885149842, 4.90853663868747665439950088080, 5.86207681602135082004995696436, 7.38359355245444053310585557251, 7.949288001457336765291419684749, 8.621373122572717417447165366548, 9.541503303718091858844022581433

Graph of the ZZ-function along the critical line