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Properties

Label 2-9282-1.1-c1-0-146
Degree 22
Conductor 92829282
Sign 11
Analytic cond. 74.117174.1171
Root an. cond. 8.609138.60913
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 3-s + 4-s + 4·5-s + 6-s + 7-s + 8-s + 9-s + 4·10-s + 4·11-s + 12-s − 13-s + 14-s + 4·15-s + 16-s − 17-s + 18-s + 4·20-s + 21-s + 4·22-s + 4·23-s + 24-s + 11·25-s − 26-s + 27-s + 28-s + 4·30-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 1/2·4-s + 1.78·5-s + 0.408·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s + 1.26·10-s + 1.20·11-s + 0.288·12-s − 0.277·13-s + 0.267·14-s + 1.03·15-s + 1/4·16-s − 0.242·17-s + 0.235·18-s + 0.894·20-s + 0.218·21-s + 0.852·22-s + 0.834·23-s + 0.204·24-s + 11/5·25-s − 0.196·26-s + 0.192·27-s + 0.188·28-s + 0.730·30-s + ⋯

Functional equation

Λ(s)=(9282s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 9282 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(9282s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 9282 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 92829282    =    2⋅3⋅7⋅13⋅172 \cdot 3 \cdot 7 \cdot 13 \cdot 17
Sign: 11
Analytic conductor: 74.117174.1171
Root analytic conductor: 8.609138.60913
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 9282, ( :1/2), 1)(2,\ 9282,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 7.1757845457.175784545
L(12)L(\frac12) ≈\approx 7.1757845457.175784545
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
3 1−T 1 - T
7 1−T 1 - T
13 1+T 1 + T
17 1+T 1 + T
good5 1−4T+pT2 1 - 4 T + p T^{2}
11 1−4T+pT2 1 - 4 T + p T^{2}
19 1+pT2 1 + p T^{2}
23 1−4T+pT2 1 - 4 T + p T^{2}
29 1+pT2 1 + p T^{2}
31 1+8T+pT2 1 + 8 T + p T^{2}
37 1+8T+pT2 1 + 8 T + p T^{2}
41 1−4T+pT2 1 - 4 T + p T^{2}
43 1+4T+pT2 1 + 4 T + p T^{2}
47 1−6T+pT2 1 - 6 T + p T^{2}
53 1−2T+pT2 1 - 2 T + p T^{2}
59 1+10T+pT2 1 + 10 T + p T^{2}
61 1−8T+pT2 1 - 8 T + p T^{2}
67 1+2T+pT2 1 + 2 T + p T^{2}
71 1+pT2 1 + p T^{2}
73 1+14T+pT2 1 + 14 T + p T^{2}
79 1+pT2 1 + p T^{2}
83 1−10T+pT2 1 - 10 T + p T^{2}
89 1+10T+pT2 1 + 10 T + p T^{2}
97 1+14T+pT2 1 + 14 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

−7.43431622225293476980256076857, −6.91485885108834120445811403430, −6.30002641476588263934450780620, −5.58752905368971229950122030594, −5.05216499506929163214392185002, −4.23639732023900214253770110943, −3.39905678903158564906056117992, −2.57135603277719859180202295969, −1.83970930829277438866929754733, −1.29631681266387487357538053133, 1.29631681266387487357538053133, 1.83970930829277438866929754733, 2.57135603277719859180202295969, 3.39905678903158564906056117992, 4.23639732023900214253770110943, 5.05216499506929163214392185002, 5.58752905368971229950122030594, 6.30002641476588263934450780620, 6.91485885108834120445811403430, 7.43431622225293476980256076857

Graph of the ZZ-function along the critical line