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Properties

Label 2-4001-1.1-c1-0-1
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  − 0.114·2-s + 0.742·3-s − 1.98·4-s − 1.13·5-s − 0.0853·6-s − 2.76·7-s + 0.458·8-s − 2.44·9-s + 0.130·10-s − 5.26·11-s − 1.47·12-s − 5.84·13-s + 0.317·14-s − 0.841·15-s + 3.92·16-s − 0.422·17-s + 0.281·18-s − 0.824·19-s + 2.25·20-s − 2.04·21-s + 0.605·22-s + 3.44·23-s + 0.340·24-s − 3.71·25-s + 0.671·26-s − 4.04·27-s + 5.48·28-s + ⋯
L(s)  = 1  − 0.0813·2-s + 0.428·3-s − 0.993·4-s − 0.506·5-s − 0.0348·6-s − 1.04·7-s + 0.162·8-s − 0.816·9-s + 0.0412·10-s − 1.58·11-s − 0.425·12-s − 1.61·13-s + 0.0848·14-s − 0.217·15-s + 0.980·16-s − 0.102·17-s + 0.0663·18-s − 0.189·19-s + 0.503·20-s − 0.447·21-s + 0.129·22-s + 0.717·23-s + 0.0694·24-s − 0.742·25-s + 0.131·26-s − 0.778·27-s + 1.03·28-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 0.00089454800700.0008945480070
L(12)L(\frac12) ≈\approx 0.00089454800700.0008945480070
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+0.114T+2T2 1 + 0.114T + 2T^{2}
3 1−0.742T+3T2 1 - 0.742T + 3T^{2}
5 1+1.13T+5T2 1 + 1.13T + 5T^{2}
7 1+2.76T+7T2 1 + 2.76T + 7T^{2}
11 1+5.26T+11T2 1 + 5.26T + 11T^{2}
13 1+5.84T+13T2 1 + 5.84T + 13T^{2}
17 1+0.422T+17T2 1 + 0.422T + 17T^{2}
19 1+0.824T+19T2 1 + 0.824T + 19T^{2}
23 1−3.44T+23T2 1 - 3.44T + 23T^{2}
29 1+5.32T+29T2 1 + 5.32T + 29T^{2}
31 1−2.65T+31T2 1 - 2.65T + 31T^{2}
37 1+6.68T+37T2 1 + 6.68T + 37T^{2}
41 1+3.98T+41T2 1 + 3.98T + 41T^{2}
43 1+12.5T+43T2 1 + 12.5T + 43T^{2}
47 1−3.54T+47T2 1 - 3.54T + 47T^{2}
53 1+5.28T+53T2 1 + 5.28T + 53T^{2}
59 1+0.983T+59T2 1 + 0.983T + 59T^{2}
61 1−1.72T+61T2 1 - 1.72T + 61T^{2}
67 1−2.77T+67T2 1 - 2.77T + 67T^{2}
71 1+11.2T+71T2 1 + 11.2T + 71T^{2}
73 1−5.78T+73T2 1 - 5.78T + 73T^{2}
79 1−0.287T+79T2 1 - 0.287T + 79T^{2}
83 1−2.08T+83T2 1 - 2.08T + 83T^{2}
89 1−12.1T+89T2 1 - 12.1T + 89T^{2}
97 1+1.82T+97T2 1 + 1.82T + 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

−8.496247004743205087455002710072, −7.75499968942466468648199861878, −7.30641735549044522108677949865, −6.17668816045243159582701043218, −5.18376303959556518553405653475, −4.89372586374907577249243822460, −3.62502485894195735959023040756, −3.10225480066630781210394258874, −2.22102103875291069700402747212, −0.01561392479398394160029485851, 0.01561392479398394160029485851, 2.22102103875291069700402747212, 3.10225480066630781210394258874, 3.62502485894195735959023040756, 4.89372586374907577249243822460, 5.18376303959556518553405653475, 6.17668816045243159582701043218, 7.30641735549044522108677949865, 7.75499968942466468648199861878, 8.496247004743205087455002710072

Graph of the ZZ-function along the critical line