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Properties

Label 2-4000-1.1-c1-0-37
Degree 22
Conductor 40004000
Sign 11
Analytic cond. 31.940131.9401
Root an. cond. 5.651565.65156
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  + 2.40·3-s − 2.20·7-s + 2.78·9-s + 1.68·11-s + 6.87·13-s − 5.86·17-s + 0.108·19-s − 5.31·21-s + 4.40·23-s − 0.513·27-s − 4.25·29-s + 9.07·31-s + 4.04·33-s + 1.74·37-s + 16.5·39-s + 7.95·41-s − 8.89·43-s + 10.5·47-s − 2.12·49-s − 14.1·51-s + 8.51·53-s + 0.260·57-s + 7.67·59-s + 1.94·61-s − 6.15·63-s + 0.788·67-s + 10.5·69-s + ⋯
L(s)  = 1  + 1.38·3-s − 0.834·7-s + 0.928·9-s + 0.507·11-s + 1.90·13-s − 1.42·17-s + 0.0248·19-s − 1.15·21-s + 0.918·23-s − 0.0987·27-s − 0.790·29-s + 1.62·31-s + 0.704·33-s + 0.287·37-s + 2.64·39-s + 1.24·41-s − 1.35·43-s + 1.53·47-s − 0.302·49-s − 1.97·51-s + 1.16·53-s + 0.0345·57-s + 0.999·59-s + 0.249·61-s − 0.775·63-s + 0.0963·67-s + 1.27·69-s + ⋯

Functional equation

Λ(s)=(4000s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(4000s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 40004000    =    25⋅532^{5} \cdot 5^{3}
Sign: 11
Analytic conductor: 31.940131.9401
Root analytic conductor: 5.651565.65156
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 4000, ( :1/2), 1)(2,\ 4000,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 3.2240666103.224066610
L(12)L(\frac12) ≈\approx 3.2240666103.224066610
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
good3 1−2.40T+3T2 1 - 2.40T + 3T^{2}
7 1+2.20T+7T2 1 + 2.20T + 7T^{2}
11 1−1.68T+11T2 1 - 1.68T + 11T^{2}
13 1−6.87T+13T2 1 - 6.87T + 13T^{2}
17 1+5.86T+17T2 1 + 5.86T + 17T^{2}
19 1−0.108T+19T2 1 - 0.108T + 19T^{2}
23 1−4.40T+23T2 1 - 4.40T + 23T^{2}
29 1+4.25T+29T2 1 + 4.25T + 29T^{2}
31 1−9.07T+31T2 1 - 9.07T + 31T^{2}
37 1−1.74T+37T2 1 - 1.74T + 37T^{2}
41 1−7.95T+41T2 1 - 7.95T + 41T^{2}
43 1+8.89T+43T2 1 + 8.89T + 43T^{2}
47 1−10.5T+47T2 1 - 10.5T + 47T^{2}
53 1−8.51T+53T2 1 - 8.51T + 53T^{2}
59 1−7.67T+59T2 1 - 7.67T + 59T^{2}
61 1−1.94T+61T2 1 - 1.94T + 61T^{2}
67 1−0.788T+67T2 1 - 0.788T + 67T^{2}
71 1−5.48T+71T2 1 - 5.48T + 71T^{2}
73 1−11.9T+73T2 1 - 11.9T + 73T^{2}
79 1−1.28T+79T2 1 - 1.28T + 79T^{2}
83 1+4.08T+83T2 1 + 4.08T + 83T^{2}
89 1+2.37T+89T2 1 + 2.37T + 89T^{2}
97 1−5.70T+97T2 1 - 5.70T + 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

−8.534711056562788620986556491733, −7.994161633014397889341414554312, −6.85835220760440089928627928690, −6.51657821095068523070446606849, −5.58814440738750453402582690095, −4.24018134683575526941493658189, −3.77116765835920442107650954761, −2.98690796114780750824466484933, −2.19942232436056688573320124342, −1.00066617030322062623973204524, 1.00066617030322062623973204524, 2.19942232436056688573320124342, 2.98690796114780750824466484933, 3.77116765835920442107650954761, 4.24018134683575526941493658189, 5.58814440738750453402582690095, 6.51657821095068523070446606849, 6.85835220760440089928627928690, 7.994161633014397889341414554312, 8.534711056562788620986556491733

Graph of the ZZ-function along the critical line