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Properties

Label 2-350-1.1-c1-0-7
Degree 22
Conductor 350350
Sign −1-1
Analytic cond. 2.794762.79476
Root an. cond. 1.671751.67175
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 3-s + 4-s + 6-s − 7-s − 8-s − 2·9-s + 3·11-s − 12-s − 2·13-s + 14-s + 16-s − 3·17-s + 2·18-s − 7·19-s + 21-s − 3·22-s + 24-s + 2·26-s + 5·27-s − 28-s − 6·29-s − 4·31-s − 32-s − 3·33-s + 3·34-s − 2·36-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.377·7-s − 0.353·8-s − 2/3·9-s + 0.904·11-s − 0.288·12-s − 0.554·13-s + 0.267·14-s + 1/4·16-s − 0.727·17-s + 0.471·18-s − 1.60·19-s + 0.218·21-s − 0.639·22-s + 0.204·24-s + 0.392·26-s + 0.962·27-s − 0.188·28-s − 1.11·29-s − 0.718·31-s − 0.176·32-s − 0.522·33-s + 0.514·34-s − 1/3·36-s + ⋯

Functional equation

Λ(s)=(350s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(350s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, 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: 2.794762.79476
Root analytic conductor: 1.671751.67175
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 350, ( :1/2), −1)(2,\ 350,\ (\ :1/2),\ -1)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
L(32)L(\frac{3}{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+T 1 + T
5 1 1
7 1+T 1 + T
good3 1+T+pT2 1 + T + p T^{2}
11 1−3T+pT2 1 - 3 T + p T^{2}
13 1+2T+pT2 1 + 2 T + p T^{2}
17 1+3T+pT2 1 + 3 T + p T^{2}
19 1+7T+pT2 1 + 7 T + p T^{2}
23 1+pT2 1 + p T^{2}
29 1+6T+pT2 1 + 6 T + p T^{2}
31 1+4T+pT2 1 + 4 T + p T^{2}
37 1+8T+pT2 1 + 8 T + p T^{2}
41 1+9T+pT2 1 + 9 T + p T^{2}
43 1+8T+pT2 1 + 8 T + p T^{2}
47 1−6T+pT2 1 - 6 T + p T^{2}
53 1−12T+pT2 1 - 12 T + p T^{2}
59 1−12T+pT2 1 - 12 T + p T^{2}
61 1+10T+pT2 1 + 10 T + p T^{2}
67 1−7T+pT2 1 - 7 T + p T^{2}
71 1−6T+pT2 1 - 6 T + p T^{2}
73 1+5T+pT2 1 + 5 T + p T^{2}
79 1−14T+pT2 1 - 14 T + p T^{2}
83 1−9T+pT2 1 - 9 T + p T^{2}
89 1+15T+pT2 1 + 15 T + p T^{2}
97 1−10T+pT2 1 - 10 T + p 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

−10.97618644248142866152725079598, −10.18616620126561124117671237886, −9.056880838216350511563756479245, −8.489865387400252022087843283105, −7.02864082573576697328686349834, −6.40152272675796392235796567725, −5.26770803610804114861303773339, −3.76060769659553506276694871497, −2.12269688406492296074462119899, 0, 2.12269688406492296074462119899, 3.76060769659553506276694871497, 5.26770803610804114861303773339, 6.40152272675796392235796567725, 7.02864082573576697328686349834, 8.489865387400252022087843283105, 9.056880838216350511563756479245, 10.18616620126561124117671237886, 10.97618644248142866152725079598

Graph of the ZZ-function along the critical line