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Properties

Label 2-504-1.1-c7-0-43
Degree 22
Conductor 504504
Sign −1-1
Analytic cond. 157.442157.442
Root an. cond. 12.547512.5475
Motivic weight 77
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 160·5-s − 343·7-s + 6.84e3·11-s − 2.90e3·13-s − 1.65e4·17-s − 6.71e3·19-s + 976·23-s − 5.25e4·25-s + 6.16e4·29-s − 6.92e4·31-s − 5.48e4·35-s − 5.33e5·37-s − 1.83e5·41-s + 9.66e5·43-s + 1.90e5·47-s + 1.17e5·49-s + 7.85e5·53-s + 1.09e6·55-s − 2.89e6·59-s − 9.58e4·61-s − 4.64e5·65-s − 9.91e5·67-s − 1.06e6·71-s + 2.52e6·73-s − 2.34e6·77-s + 2.85e5·79-s − 7.09e6·83-s + ⋯
L(s)  = 1  + 0.572·5-s − 0.377·7-s + 1.54·11-s − 0.366·13-s − 0.817·17-s − 0.224·19-s + 0.0167·23-s − 0.672·25-s + 0.469·29-s − 0.417·31-s − 0.216·35-s − 1.73·37-s − 0.415·41-s + 1.85·43-s + 0.267·47-s + 1/7·49-s + 0.724·53-s + 0.886·55-s − 1.83·59-s − 0.0540·61-s − 0.209·65-s − 0.402·67-s − 0.354·71-s + 0.759·73-s − 0.585·77-s + 0.0652·79-s − 1.36·83-s + ⋯

Functional equation

Λ(s)=(504s/2 ΓC(s) L(s)=(− Λ(8−s)\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}
Λ(s)=(504s/2 ΓC(s+7/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 504504    =    23⋅32⋅72^{3} \cdot 3^{2} \cdot 7
Sign: −1-1
Analytic conductor: 157.442157.442
Root analytic conductor: 12.547512.5475
Motivic weight: 77
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 504, ( :7/2), −1)(2,\ 504,\ (\ :7/2),\ -1)

Particular Values

L(4)L(4) == 00
L(12)L(\frac12) == 00
L(92)L(\frac{9}{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)
bad2 1 1
3 1 1
7 1+p3T 1 + p^{3} T
good5 1−32pT+p7T2 1 - 32 p T + p^{7} T^{2}
11 1−6840T+p7T2 1 - 6840 T + p^{7} T^{2}
13 1+2900T+p7T2 1 + 2900 T + p^{7} T^{2}
17 1+16566T+p7T2 1 + 16566 T + p^{7} T^{2}
19 1+6718T+p7T2 1 + 6718 T + p^{7} T^{2}
23 1−976T+p7T2 1 - 976 T + p^{7} T^{2}
29 1−61662T+p7T2 1 - 61662 T + p^{7} T^{2}
31 1+69236T+p7T2 1 + 69236 T + p^{7} T^{2}
37 1+533062T+p7T2 1 + 533062 T + p^{7} T^{2}
41 1+183158T+p7T2 1 + 183158 T + p^{7} T^{2}
43 1−966864T+p7T2 1 - 966864 T + p^{7} T^{2}
47 1−190268T+p7T2 1 - 190268 T + p^{7} T^{2}
53 1−785010T+p7T2 1 - 785010 T + p^{7} T^{2}
59 1+2893594T+p7T2 1 + 2893594 T + p^{7} T^{2}
61 1+95896T+p7T2 1 + 95896 T + p^{7} T^{2}
67 1+991644T+p7T2 1 + 991644 T + p^{7} T^{2}
71 1+1068160T+p7T2 1 + 1068160 T + p^{7} T^{2}
73 1−2523458T+p7T2 1 - 2523458 T + p^{7} T^{2}
79 1−285848T+p7T2 1 - 285848 T + p^{7} T^{2}
83 1+7094938T+p7T2 1 + 7094938 T + p^{7} T^{2}
89 1−252390T+p7T2 1 - 252390 T + p^{7} T^{2}
97 1+1824794T+p7T2 1 + 1824794 T + p^{7} 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

−9.258613466510256434370301669858, −8.733258954856603251660315078854, −7.33697557130672408377368667970, −6.53872848419034886629005382822, −5.78166006549148570393319928509, −4.52099969654228695501655265317, −3.59951770344855555979841821508, −2.30583993716523161450841664353, −1.33801742610716765808418334057, 0, 1.33801742610716765808418334057, 2.30583993716523161450841664353, 3.59951770344855555979841821508, 4.52099969654228695501655265317, 5.78166006549148570393319928509, 6.53872848419034886629005382822, 7.33697557130672408377368667970, 8.733258954856603251660315078854, 9.258613466510256434370301669858

Graph of the ZZ-function along the critical line