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Properties

Label 2-4001-1.1-c1-0-122
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  + 2.45·2-s − 1.68·3-s + 4.02·4-s − 3.39·5-s − 4.13·6-s + 4.85·7-s + 4.97·8-s − 0.165·9-s − 8.32·10-s − 3.48·11-s − 6.78·12-s − 0.426·13-s + 11.9·14-s + 5.71·15-s + 4.16·16-s − 4.22·17-s − 0.405·18-s + 7.37·19-s − 13.6·20-s − 8.17·21-s − 8.54·22-s + 5.74·23-s − 8.38·24-s + 6.50·25-s − 1.04·26-s + 5.32·27-s + 19.5·28-s + ⋯
L(s)  = 1  + 1.73·2-s − 0.972·3-s + 2.01·4-s − 1.51·5-s − 1.68·6-s + 1.83·7-s + 1.76·8-s − 0.0550·9-s − 2.63·10-s − 1.04·11-s − 1.95·12-s − 0.118·13-s + 3.18·14-s + 1.47·15-s + 1.04·16-s − 1.02·17-s − 0.0955·18-s + 1.69·19-s − 3.05·20-s − 1.78·21-s − 1.82·22-s + 1.19·23-s − 1.71·24-s + 1.30·25-s − 0.205·26-s + 1.02·27-s + 3.69·28-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 3.3189236833.318923683
L(12)L(\frac12) ≈\approx 3.3189236833.318923683
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 1−2.45T+2T2 1 - 2.45T + 2T^{2}
3 1+1.68T+3T2 1 + 1.68T + 3T^{2}
5 1+3.39T+5T2 1 + 3.39T + 5T^{2}
7 1−4.85T+7T2 1 - 4.85T + 7T^{2}
11 1+3.48T+11T2 1 + 3.48T + 11T^{2}
13 1+0.426T+13T2 1 + 0.426T + 13T^{2}
17 1+4.22T+17T2 1 + 4.22T + 17T^{2}
19 1−7.37T+19T2 1 - 7.37T + 19T^{2}
23 1−5.74T+23T2 1 - 5.74T + 23T^{2}
29 1+1.35T+29T2 1 + 1.35T + 29T^{2}
31 1−1.93T+31T2 1 - 1.93T + 31T^{2}
37 1−5.57T+37T2 1 - 5.57T + 37T^{2}
41 1+3.32T+41T2 1 + 3.32T + 41T^{2}
43 1+3.08T+43T2 1 + 3.08T + 43T^{2}
47 1−13.0T+47T2 1 - 13.0T + 47T^{2}
53 1−2.68T+53T2 1 - 2.68T + 53T^{2}
59 1+7.51T+59T2 1 + 7.51T + 59T^{2}
61 1−13.7T+61T2 1 - 13.7T + 61T^{2}
67 1+4.51T+67T2 1 + 4.51T + 67T^{2}
71 1+2.97T+71T2 1 + 2.97T + 71T^{2}
73 1−13.5T+73T2 1 - 13.5T + 73T^{2}
79 1+11.0T+79T2 1 + 11.0T + 79T^{2}
83 1−10.2T+83T2 1 - 10.2T + 83T^{2}
89 1−3.18T+89T2 1 - 3.18T + 89T^{2}
97 1−10.7T+97T2 1 - 10.7T + 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

−8.067991564242208936993556816381, −7.45546299340662686647666872355, −6.97186395418229750170548874309, −5.80939453328498079209743412057, −5.12490886197908036609856640196, −4.85237701486682776393366080401, −4.21377060569544978828938507973, −3.22509505840709457463816239884, −2.35041701963252123403757317829, −0.848266441908852189780975776656, 0.848266441908852189780975776656, 2.35041701963252123403757317829, 3.22509505840709457463816239884, 4.21377060569544978828938507973, 4.85237701486682776393366080401, 5.12490886197908036609856640196, 5.80939453328498079209743412057, 6.97186395418229750170548874309, 7.45546299340662686647666872355, 8.067991564242208936993556816381

Graph of the ZZ-function along the critical line