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Properties

Label 2-4001-1.1-c1-0-135
Degree 22
Conductor 40014001
Sign 11
Analytic cond. 31.948131.9481
Root an. cond. 5.652265.65226
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  + 0.799·2-s − 0.343·3-s − 1.36·4-s + 3.87·5-s − 0.274·6-s + 3.67·7-s − 2.68·8-s − 2.88·9-s + 3.09·10-s + 3.19·11-s + 0.467·12-s − 5.30·13-s + 2.93·14-s − 1.33·15-s + 0.575·16-s − 1.86·17-s − 2.30·18-s − 2.90·19-s − 5.27·20-s − 1.26·21-s + 2.55·22-s − 3.28·23-s + 0.923·24-s + 10.0·25-s − 4.23·26-s + 2.02·27-s − 4.99·28-s + ⋯
L(s)  = 1  + 0.565·2-s − 0.198·3-s − 0.680·4-s + 1.73·5-s − 0.112·6-s + 1.38·7-s − 0.949·8-s − 0.960·9-s + 0.979·10-s + 0.962·11-s + 0.135·12-s − 1.47·13-s + 0.783·14-s − 0.343·15-s + 0.143·16-s − 0.453·17-s − 0.542·18-s − 0.666·19-s − 1.17·20-s − 0.275·21-s + 0.543·22-s − 0.684·23-s + 0.188·24-s + 2.00·25-s − 0.831·26-s + 0.389·27-s − 0.944·28-s + ⋯

Functional equation

Λ(s)=(4001s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 4001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(4001s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 4001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 40014001
Sign: 11
Analytic conductor: 31.948131.9481
Root analytic conductor: 5.652265.65226
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 4001, ( :1/2), 1)(2,\ 4001,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.8913588652.891358865
L(12)L(\frac12) ≈\approx 2.8913588652.891358865
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)
bad4001 1+O(T) 1+O(T)
good2 1−0.799T+2T2 1 - 0.799T + 2T^{2}
3 1+0.343T+3T2 1 + 0.343T + 3T^{2}
5 1−3.87T+5T2 1 - 3.87T + 5T^{2}
7 1−3.67T+7T2 1 - 3.67T + 7T^{2}
11 1−3.19T+11T2 1 - 3.19T + 11T^{2}
13 1+5.30T+13T2 1 + 5.30T + 13T^{2}
17 1+1.86T+17T2 1 + 1.86T + 17T^{2}
19 1+2.90T+19T2 1 + 2.90T + 19T^{2}
23 1+3.28T+23T2 1 + 3.28T + 23T^{2}
29 1−8.38T+29T2 1 - 8.38T + 29T^{2}
31 1−1.37T+31T2 1 - 1.37T + 31T^{2}
37 1−10.9T+37T2 1 - 10.9T + 37T^{2}
41 1+5.18T+41T2 1 + 5.18T + 41T^{2}
43 1−9.49T+43T2 1 - 9.49T + 43T^{2}
47 1−8.49T+47T2 1 - 8.49T + 47T^{2}
53 1−11.4T+53T2 1 - 11.4T + 53T^{2}
59 1−6.65T+59T2 1 - 6.65T + 59T^{2}
61 1+5.18T+61T2 1 + 5.18T + 61T^{2}
67 1+3.20T+67T2 1 + 3.20T + 67T^{2}
71 1+5.41T+71T2 1 + 5.41T + 71T^{2}
73 1−5.64T+73T2 1 - 5.64T + 73T^{2}
79 1+7.91T+79T2 1 + 7.91T + 79T^{2}
83 1−0.854T+83T2 1 - 0.854T + 83T^{2}
89 1+8.87T+89T2 1 + 8.87T + 89T^{2}
97 1−6.76T+97T2 1 - 6.76T + 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

−8.780073311396209564318381208603, −7.80505573000856290891969715068, −6.65360441435251370153750653804, −5.97234235687817244356235595963, −5.47690737362185499590690517674, −4.70521529424060465381552617697, −4.27445532800415361448489526314, −2.68335254251592676766043791362, −2.21002558953799022561149195323, −0.933658214600414393494774710756, 0.933658214600414393494774710756, 2.21002558953799022561149195323, 2.68335254251592676766043791362, 4.27445532800415361448489526314, 4.70521529424060465381552617697, 5.47690737362185499590690517674, 5.97234235687817244356235595963, 6.65360441435251370153750653804, 7.80505573000856290891969715068, 8.780073311396209564318381208603

Graph of the ZZ-function along the critical line