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Properties

Label 2-693-1.1-c3-0-35
Degree 22
Conductor 693693
Sign −1-1
Analytic cond. 40.888340.8883
Root an. cond. 6.394396.39439
Motivic weight 33
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s − 4·4-s − 11·5-s − 7·7-s + 24·8-s + 22·10-s − 11·11-s − 5·13-s + 14·14-s − 16·16-s + 118·17-s − 105·19-s + 44·20-s + 22·22-s + 68·23-s − 4·25-s + 10·26-s + 28·28-s + 195·29-s + 214·31-s − 160·32-s − 236·34-s + 77·35-s + 33·37-s + 210·38-s − 264·40-s + 376·41-s + ⋯
L(s)  = 1  − 0.707·2-s − 1/2·4-s − 0.983·5-s − 0.377·7-s + 1.06·8-s + 0.695·10-s − 0.301·11-s − 0.106·13-s + 0.267·14-s − 1/4·16-s + 1.68·17-s − 1.26·19-s + 0.491·20-s + 0.213·22-s + 0.616·23-s − 0.0319·25-s + 0.0754·26-s + 0.188·28-s + 1.24·29-s + 1.23·31-s − 0.883·32-s − 1.19·34-s + 0.371·35-s + 0.146·37-s + 0.896·38-s − 1.04·40-s + 1.43·41-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: yes
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+pT 1 + p T
11 1+pT 1 + p T
good2 1+pT+p3T2 1 + p T + p^{3} T^{2}
5 1+11T+p3T2 1 + 11 T + p^{3} T^{2}
13 1+5T+p3T2 1 + 5 T + p^{3} T^{2}
17 1−118T+p3T2 1 - 118 T + p^{3} T^{2}
19 1+105T+p3T2 1 + 105 T + p^{3} T^{2}
23 1−68T+p3T2 1 - 68 T + p^{3} T^{2}
29 1−195T+p3T2 1 - 195 T + p^{3} T^{2}
31 1−214T+p3T2 1 - 214 T + p^{3} T^{2}
37 1−33T+p3T2 1 - 33 T + p^{3} T^{2}
41 1−376T+p3T2 1 - 376 T + p^{3} T^{2}
43 1+168T+p3T2 1 + 168 T + p^{3} T^{2}
47 1+61T+p3T2 1 + 61 T + p^{3} T^{2}
53 1+24T+p3T2 1 + 24 T + p^{3} T^{2}
59 1+625T+p3T2 1 + 625 T + p^{3} T^{2}
61 1+558T+p3T2 1 + 558 T + p^{3} T^{2}
67 1−173T+p3T2 1 - 173 T + p^{3} T^{2}
71 1+168T+p3T2 1 + 168 T + p^{3} T^{2}
73 1−973T+p3T2 1 - 973 T + p^{3} T^{2}
79 1+1072T+p3T2 1 + 1072 T + p^{3} T^{2}
83 1+1458T+p3T2 1 + 1458 T + p^{3} T^{2}
89 1−198T+p3T2 1 - 198 T + p^{3} T^{2}
97 1+352T+p3T2 1 + 352 T + p^{3} T^{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

−9.690080354283100428463328679529, −8.630725087549140185856852816193, −8.029033511105383098361807536901, −7.36376097778730294582595768946, −6.16131591365869911538356879012, −4.87520469019249353765328428154, −4.06318388236473030531951589967, −2.92755015703034847216578885872, −1.10133640584924975490442967894, 0, 1.10133640584924975490442967894, 2.92755015703034847216578885872, 4.06318388236473030531951589967, 4.87520469019249353765328428154, 6.16131591365869911538356879012, 7.36376097778730294582595768946, 8.029033511105383098361807536901, 8.630725087549140185856852816193, 9.690080354283100428463328679529

Graph of the ZZ-function along the critical line