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Properties

Label 2-4002-1.1-c1-0-14
Degree 22
Conductor 40024002
Sign 11
Analytic cond. 31.956131.9561
Root an. cond. 5.652975.65297
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-s − 3-s + 4-s − 1.12·5-s − 6-s − 0.350·7-s + 8-s + 9-s − 1.12·10-s − 2.98·11-s − 12-s + 2.39·13-s − 0.350·14-s + 1.12·15-s + 16-s − 4.11·17-s + 18-s + 8.36·19-s − 1.12·20-s + 0.350·21-s − 2.98·22-s − 23-s − 24-s − 3.73·25-s + 2.39·26-s − 27-s − 0.350·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.503·5-s − 0.408·6-s − 0.132·7-s + 0.353·8-s + 0.333·9-s − 0.356·10-s − 0.901·11-s − 0.288·12-s + 0.664·13-s − 0.0936·14-s + 0.290·15-s + 0.250·16-s − 0.998·17-s + 0.235·18-s + 1.91·19-s − 0.251·20-s + 0.0764·21-s − 0.637·22-s − 0.208·23-s − 0.204·24-s − 0.746·25-s + 0.469·26-s − 0.192·27-s − 0.0662·28-s + ⋯

Functional equation

Λ(s)=(4002s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 4002 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(4002s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 4002 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 40024002    =    2⋅3⋅23⋅292 \cdot 3 \cdot 23 \cdot 29
Sign: 11
Analytic conductor: 31.956131.9561
Root analytic conductor: 5.652975.65297
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 4002, ( :1/2), 1)(2,\ 4002,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.0117875132.011787513
L(12)L(\frac12) ≈\approx 2.0117875132.011787513
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−T 1 - T
3 1+T 1 + T
23 1+T 1 + T
29 1+T 1 + T
good5 1+1.12T+5T2 1 + 1.12T + 5T^{2}
7 1+0.350T+7T2 1 + 0.350T + 7T^{2}
11 1+2.98T+11T2 1 + 2.98T + 11T^{2}
13 1−2.39T+13T2 1 - 2.39T + 13T^{2}
17 1+4.11T+17T2 1 + 4.11T + 17T^{2}
19 1−8.36T+19T2 1 - 8.36T + 19T^{2}
31 1−7.08T+31T2 1 - 7.08T + 31T^{2}
37 1+8.11T+37T2 1 + 8.11T + 37T^{2}
41 1−8.02T+41T2 1 - 8.02T + 41T^{2}
43 1−5.37T+43T2 1 - 5.37T + 43T^{2}
47 1+1.42T+47T2 1 + 1.42T + 47T^{2}
53 1−1.66T+53T2 1 - 1.66T + 53T^{2}
59 1+12.0T+59T2 1 + 12.0T + 59T^{2}
61 1−4.98T+61T2 1 - 4.98T + 61T^{2}
67 1−13.7T+67T2 1 - 13.7T + 67T^{2}
71 1−4.56T+71T2 1 - 4.56T + 71T^{2}
73 1−16.5T+73T2 1 - 16.5T + 73T^{2}
79 1−12.4T+79T2 1 - 12.4T + 79T^{2}
83 1−1.75T+83T2 1 - 1.75T + 83T^{2}
89 1+7.49T+89T2 1 + 7.49T + 89T^{2}
97 1−14.1T+97T2 1 - 14.1T + 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

−8.139082452161990939872671744612, −7.68031456720024839113893264573, −6.84537093591514003515926713445, −6.14894085166250891529018400462, −5.37896614293802357743846322803, −4.78958923365078245463753222779, −3.88789230079512920320002557060, −3.16492370049393667878960154546, −2.10351093880690011808811823647, −0.74554119185062501919586894212, 0.74554119185062501919586894212, 2.10351093880690011808811823647, 3.16492370049393667878960154546, 3.88789230079512920320002557060, 4.78958923365078245463753222779, 5.37896614293802357743846322803, 6.14894085166250891529018400462, 6.84537093591514003515926713445, 7.68031456720024839113893264573, 8.139082452161990939872671744612

Graph of the ZZ-function along the critical line