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Properties

Label 2-4600-1.1-c1-0-3
Degree 22
Conductor 46004600
Sign 11
Analytic cond. 36.731136.7311
Root an. cond. 6.060626.06062
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.801·3-s − 1.60·7-s − 2.35·9-s − 6.09·11-s − 1.35·13-s − 1.60·17-s + 1.78·19-s + 1.28·21-s − 23-s + 4.29·27-s − 4.82·29-s + 6.15·31-s + 4.89·33-s − 9.87·37-s + 1.08·39-s − 8.89·41-s − 11.3·43-s + 9.85·47-s − 4.42·49-s + 1.28·51-s + 9.20·53-s − 1.42·57-s − 7.27·59-s + 0.933·61-s + 3.78·63-s + 9.42·67-s + 0.801·69-s + ⋯
L(s)  = 1  − 0.462·3-s − 0.606·7-s − 0.785·9-s − 1.83·11-s − 0.376·13-s − 0.388·17-s + 0.408·19-s + 0.280·21-s − 0.208·23-s + 0.826·27-s − 0.896·29-s + 1.10·31-s + 0.851·33-s − 1.62·37-s + 0.174·39-s − 1.38·41-s − 1.73·43-s + 1.43·47-s − 0.632·49-s + 0.180·51-s + 1.26·53-s − 0.189·57-s − 0.947·59-s + 0.119·61-s + 0.476·63-s + 1.15·67-s + 0.0965·69-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: no
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 0.50002399310.5000239931
L(12)L(\frac12) ≈\approx 0.50002399310.5000239931
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+0.801T+3T2 1 + 0.801T + 3T^{2}
7 1+1.60T+7T2 1 + 1.60T + 7T^{2}
11 1+6.09T+11T2 1 + 6.09T + 11T^{2}
13 1+1.35T+13T2 1 + 1.35T + 13T^{2}
17 1+1.60T+17T2 1 + 1.60T + 17T^{2}
19 1−1.78T+19T2 1 - 1.78T + 19T^{2}
29 1+4.82T+29T2 1 + 4.82T + 29T^{2}
31 1−6.15T+31T2 1 - 6.15T + 31T^{2}
37 1+9.87T+37T2 1 + 9.87T + 37T^{2}
41 1+8.89T+41T2 1 + 8.89T + 41T^{2}
43 1+11.3T+43T2 1 + 11.3T + 43T^{2}
47 1−9.85T+47T2 1 - 9.85T + 47T^{2}
53 1−9.20T+53T2 1 - 9.20T + 53T^{2}
59 1+7.27T+59T2 1 + 7.27T + 59T^{2}
61 1−0.933T+61T2 1 - 0.933T + 61T^{2}
67 1−9.42T+67T2 1 - 9.42T + 67T^{2}
71 1−12.6T+71T2 1 - 12.6T + 71T^{2}
73 1+2.60T+73T2 1 + 2.60T + 73T^{2}
79 1+5.16T+79T2 1 + 5.16T + 79T^{2}
83 1−15.4T+83T2 1 - 15.4T + 83T^{2}
89 1+15.7T+89T2 1 + 15.7T + 89T^{2}
97 1−12.8T+97T2 1 - 12.8T + 97T^{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

−8.336766510640660917209872405795, −7.58780996156104476696554756543, −6.82410777421114822804931027757, −6.09977662577905792800304337577, −5.16508678551875178399171338200, −5.05831940585531226117667534419, −3.61740725387602433086897239574, −2.89399246098083307640500674180, −2.08547375876196695026451774694, −0.37328064475887228071451918765, 0.37328064475887228071451918765, 2.08547375876196695026451774694, 2.89399246098083307640500674180, 3.61740725387602433086897239574, 5.05831940585531226117667534419, 5.16508678551875178399171338200, 6.09977662577905792800304337577, 6.82410777421114822804931027757, 7.58780996156104476696554756543, 8.336766510640660917209872405795

Graph of the ZZ-function along the critical line