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Properties

Label 2-350-1.1-c7-0-55
Degree 22
Conductor 350350
Sign −1-1
Analytic cond. 109.334109.334
Root an. cond. 10.456310.4563
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  − 8·2-s + 63·3-s + 64·4-s − 504·6-s − 343·7-s − 512·8-s + 1.78e3·9-s − 2.72e3·11-s + 4.03e3·12-s − 5.26e3·13-s + 2.74e3·14-s + 4.09e3·16-s + 1.77e4·17-s − 1.42e4·18-s + 712·19-s − 2.16e4·21-s + 2.18e4·22-s + 2.93e4·23-s − 3.22e4·24-s + 4.21e4·26-s − 2.55e4·27-s − 2.19e4·28-s + 6.84e4·29-s + 1.85e5·31-s − 3.27e4·32-s − 1.71e5·33-s − 1.41e5·34-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.34·3-s + 1/2·4-s − 0.952·6-s − 0.377·7-s − 0.353·8-s + 0.814·9-s − 0.617·11-s + 0.673·12-s − 0.665·13-s + 0.267·14-s + 1/4·16-s + 0.873·17-s − 0.576·18-s + 0.0238·19-s − 0.509·21-s + 0.436·22-s + 0.502·23-s − 0.476·24-s + 0.470·26-s − 0.249·27-s − 0.188·28-s + 0.521·29-s + 1.11·31-s − 0.176·32-s − 0.832·33-s − 0.617·34-s + ⋯

Functional equation

Λ(s)=(350s/2 ΓC(s) L(s)=(− Λ(8−s)\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}
Λ(s)=(350s/2 ΓC(s+7/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 350350    =    2⋅52⋅72 \cdot 5^{2} \cdot 7
Sign: −1-1
Analytic conductor: 109.334109.334
Root analytic conductor: 10.456310.4563
Motivic weight: 77
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 350, ( :7/2), −1)(2,\ 350,\ (\ :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+p3T 1 + p^{3} T
5 1 1
7 1+p3T 1 + p^{3} T
good3 1−7p2T+p7T2 1 - 7 p^{2} T + p^{7} T^{2}
11 1+2727T+p7T2 1 + 2727 T + p^{7} T^{2}
13 1+5269T+p7T2 1 + 5269 T + p^{7} T^{2}
17 1−17701T+p7T2 1 - 17701 T + p^{7} T^{2}
19 1−712T+p7T2 1 - 712 T + p^{7} T^{2}
23 1−29330T+p7T2 1 - 29330 T + p^{7} T^{2}
29 1−68491T+p7T2 1 - 68491 T + p^{7} T^{2}
31 1−185026T+p7T2 1 - 185026 T + p^{7} T^{2}
37 1−6758pT+p7T2 1 - 6758 p T + p^{7} T^{2}
41 1+125814T+p7T2 1 + 125814 T + p^{7} T^{2}
43 1+747476T+p7T2 1 + 747476 T + p^{7} T^{2}
47 1+317317T+p7T2 1 + 317317 T + p^{7} T^{2}
53 1+1623246T+p7T2 1 + 1623246 T + p^{7} T^{2}
59 1+1519262T+p7T2 1 + 1519262 T + p^{7} T^{2}
61 1+54240pT+p7T2 1 + 54240 p T + p^{7} T^{2}
67 1−2272366T+p7T2 1 - 2272366 T + p^{7} T^{2}
71 1+4963104T+p7T2 1 + 4963104 T + p^{7} T^{2}
73 1+2351750T+p7T2 1 + 2351750 T + p^{7} T^{2}
79 1+2524249T+p7T2 1 + 2524249 T + p^{7} T^{2}
83 1+6051492T+p7T2 1 + 6051492 T + p^{7} T^{2}
89 1−8043880T+p7T2 1 - 8043880 T + p^{7} T^{2}
97 1−2337645T+p7T2 1 - 2337645 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

−9.744277142406022481958025357950, −8.869945153407033430357105451697, −8.020730222753910653890846736441, −7.43267196587194912865059910446, −6.22090412586055502684768309776, −4.78141411401163709592320028132, −3.23223147212647533760477151269, −2.69676092768815901449607449781, −1.45056700901309175751873594479, 0, 1.45056700901309175751873594479, 2.69676092768815901449607449781, 3.23223147212647533760477151269, 4.78141411401163709592320028132, 6.22090412586055502684768309776, 7.43267196587194912865059910446, 8.020730222753910653890846736441, 8.869945153407033430357105451697, 9.744277142406022481958025357950

Graph of the ZZ-function along the critical line