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Properties

Label 2-110670-1.1-c1-0-22
Degree 22
Conductor 110670110670
Sign 11
Analytic cond. 883.704883.704
Root an. cond. 29.727129.7271
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 + 5-s + 6-s + 7-s + 8-s + 9-s + 10-s − 4·11-s + 12-s − 2·13-s + 14-s + 15-s + 16-s + 17-s + 18-s + 4·19-s + 20-s + 21-s − 4·22-s + 8·23-s + 24-s + 25-s − 2·26-s + 27-s + 28-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s + 0.316·10-s − 1.20·11-s + 0.288·12-s − 0.554·13-s + 0.267·14-s + 0.258·15-s + 1/4·16-s + 0.242·17-s + 0.235·18-s + 0.917·19-s + 0.223·20-s + 0.218·21-s − 0.852·22-s + 1.66·23-s + 0.204·24-s + 1/5·25-s − 0.392·26-s + 0.192·27-s + 0.188·28-s + ⋯

Functional equation

Λ(s)=(110670s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 110670 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(110670s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 110670 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 110670110670    =    2⋅3⋅5⋅7⋅17⋅312 \cdot 3 \cdot 5 \cdot 7 \cdot 17 \cdot 31
Sign: 11
Analytic conductor: 883.704883.704
Root analytic conductor: 29.727129.7271
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 110670, ( :1/2), 1)(2,\ 110670,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 6.8546824306.854682430
L(12)L(\frac12) ≈\approx 6.8546824306.854682430
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
5 1−T 1 - T
7 1−T 1 - T
17 1−T 1 - T
31 1+T 1 + T
good11 1+4T+pT2 1 + 4 T + p T^{2}
13 1+2T+pT2 1 + 2 T + p T^{2}
19 1−4T+pT2 1 - 4 T + p T^{2}
23 1−8T+pT2 1 - 8 T + p T^{2}
29 1+2T+pT2 1 + 2 T + p T^{2}
37 1−6T+pT2 1 - 6 T + p T^{2}
41 1−10T+pT2 1 - 10 T + p T^{2}
43 1+4T+pT2 1 + 4 T + p T^{2}
47 1+pT2 1 + p T^{2}
53 1−6T+pT2 1 - 6 T + p T^{2}
59 1−12T+pT2 1 - 12 T + p T^{2}
61 1+2T+pT2 1 + 2 T + p T^{2}
67 1−4T+pT2 1 - 4 T + p T^{2}
71 1+8T+pT2 1 + 8 T + p T^{2}
73 1−10T+pT2 1 - 10 T + p T^{2}
79 1+pT2 1 + p T^{2}
83 1+12T+pT2 1 + 12 T + p T^{2}
89 1+6T+pT2 1 + 6 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

−13.68017142186039, −13.10464404638792, −12.85615738228401, −12.46683960089896, −11.63704726962211, −11.31662881774289, −10.73826672686368, −10.29469362412058, −9.639554702953989, −9.401891942077475, −8.620996410570280, −8.144196767658846, −7.541886544822074, −7.217238280127678, −6.742257566562262, −5.784566887400274, −5.506643723255510, −4.989996973633639, −4.505495832804927, −3.809448450756095, −3.075102800879225, −2.669893813751381, −2.243599223309768, −1.375520758646098, −0.7000837864831691, 0.7000837864831691, 1.375520758646098, 2.243599223309768, 2.669893813751381, 3.075102800879225, 3.809448450756095, 4.505495832804927, 4.989996973633639, 5.506643723255510, 5.784566887400274, 6.742257566562262, 7.217238280127678, 7.541886544822074, 8.144196767658846, 8.620996410570280, 9.401891942077475, 9.639554702953989, 10.29469362412058, 10.73826672686368, 11.31662881774289, 11.63704726962211, 12.46683960089896, 12.85615738228401, 13.10464404638792, 13.68017142186039

Graph of the ZZ-function along the critical line