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Properties

Label 2-4600-1.1-c1-0-58
Degree 22
Conductor 46004600
Sign 11
Analytic cond. 36.731136.7311
Root an. cond. 6.060626.06062
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  + 3·3-s + 2·7-s + 6·9-s − 13-s + 6·21-s − 23-s + 9·27-s − 3·29-s + 3·31-s + 8·37-s − 3·39-s + 3·41-s + 2·43-s + 11·47-s − 3·49-s + 14·53-s − 8·59-s − 4·61-s + 12·63-s + 4·67-s − 3·69-s + 7·71-s + 9·73-s + 9·81-s − 4·83-s − 9·87-s − 2·89-s + ⋯
L(s)  = 1  + 1.73·3-s + 0.755·7-s + 2·9-s − 0.277·13-s + 1.30·21-s − 0.208·23-s + 1.73·27-s − 0.557·29-s + 0.538·31-s + 1.31·37-s − 0.480·39-s + 0.468·41-s + 0.304·43-s + 1.60·47-s − 3/7·49-s + 1.92·53-s − 1.04·59-s − 0.512·61-s + 1.51·63-s + 0.488·67-s − 0.361·69-s + 0.830·71-s + 1.05·73-s + 81-s − 0.439·83-s − 0.964·87-s − 0.211·89-s + ⋯

Functional equation

Λ(s)=(4600s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(4600s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 46004600    =    23⋅52⋅232^{3} \cdot 5^{2} \cdot 23
Sign: 11
Analytic conductor: 36.731136.7311
Root analytic conductor: 6.060626.06062
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 4600, ( :1/2), 1)(2,\ 4600,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 4.3161439194.316143919
L(12)L(\frac12) ≈\approx 4.3161439194.316143919
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 1
5 1 1
23 1+T 1 + T
good3 1−pT+pT2 1 - p T + p T^{2}
7 1−2T+pT2 1 - 2 T + p T^{2}
11 1+pT2 1 + p T^{2}
13 1+T+pT2 1 + T + p T^{2}
17 1+pT2 1 + p T^{2}
19 1+pT2 1 + p T^{2}
29 1+3T+pT2 1 + 3 T + p T^{2}
31 1−3T+pT2 1 - 3 T + p T^{2}
37 1−8T+pT2 1 - 8 T + p T^{2}
41 1−3T+pT2 1 - 3 T + p T^{2}
43 1−2T+pT2 1 - 2 T + p T^{2}
47 1−11T+pT2 1 - 11 T + p T^{2}
53 1−14T+pT2 1 - 14 T + p T^{2}
59 1+8T+pT2 1 + 8 T + p T^{2}
61 1+4T+pT2 1 + 4 T + p T^{2}
67 1−4T+pT2 1 - 4 T + p T^{2}
71 1−7T+pT2 1 - 7 T + p T^{2}
73 1−9T+pT2 1 - 9 T + p T^{2}
79 1+pT2 1 + p T^{2}
83 1+4T+pT2 1 + 4 T + p T^{2}
89 1+2T+pT2 1 + 2 T + p T^{2}
97 1+18T+pT2 1 + 18 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

−8.285574047889435899285775099263, −7.72080508917808983337092272200, −7.27420693336935451926548382562, −6.24709568035458347850164186333, −5.21601831815009121569251901191, −4.30937835957950367596720771655, −3.78023121707294719609388975674, −2.71063570111023727220455874686, −2.20807718970593446330580525082, −1.14993359895366307790634424705, 1.14993359895366307790634424705, 2.20807718970593446330580525082, 2.71063570111023727220455874686, 3.78023121707294719609388975674, 4.30937835957950367596720771655, 5.21601831815009121569251901191, 6.24709568035458347850164186333, 7.27420693336935451926548382562, 7.72080508917808983337092272200, 8.285574047889435899285775099263

Graph of the ZZ-function along the critical line