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Properties

Label 2-4002-1.1-c1-0-31
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 + 1.78·5-s − 6-s + 0.126·7-s + 8-s + 9-s + 1.78·10-s − 2.47·11-s − 12-s + 4.19·13-s + 0.126·14-s − 1.78·15-s + 16-s + 7.66·17-s + 18-s + 0.126·19-s + 1.78·20-s − 0.126·21-s − 2.47·22-s + 23-s − 24-s − 1.79·25-s + 4.19·26-s − 27-s + 0.126·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.800·5-s − 0.408·6-s + 0.0478·7-s + 0.353·8-s + 0.333·9-s + 0.565·10-s − 0.745·11-s − 0.288·12-s + 1.16·13-s + 0.0338·14-s − 0.461·15-s + 0.250·16-s + 1.85·17-s + 0.235·18-s + 0.0290·19-s + 0.400·20-s − 0.0276·21-s − 0.527·22-s + 0.208·23-s − 0.204·24-s − 0.359·25-s + 0.823·26-s − 0.192·27-s + 0.0239·28-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 3.2456769253.245676925
L(12)L(\frac12) ≈\approx 3.2456769253.245676925
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 1−1.78T+5T2 1 - 1.78T + 5T^{2}
7 1−0.126T+7T2 1 - 0.126T + 7T^{2}
11 1+2.47T+11T2 1 + 2.47T + 11T^{2}
13 1−4.19T+13T2 1 - 4.19T + 13T^{2}
17 1−7.66T+17T2 1 - 7.66T + 17T^{2}
19 1−0.126T+19T2 1 - 0.126T + 19T^{2}
31 1−3.37T+31T2 1 - 3.37T + 31T^{2}
37 1−0.736T+37T2 1 - 0.736T + 37T^{2}
41 1+2.59T+41T2 1 + 2.59T + 41T^{2}
43 1+6.10T+43T2 1 + 6.10T + 43T^{2}
47 1+6.46T+47T2 1 + 6.46T + 47T^{2}
53 1−6.06T+53T2 1 - 6.06T + 53T^{2}
59 1+3.99T+59T2 1 + 3.99T + 59T^{2}
61 1−6.03T+61T2 1 - 6.03T + 61T^{2}
67 1−8.50T+67T2 1 - 8.50T + 67T^{2}
71 1−0.604T+71T2 1 - 0.604T + 71T^{2}
73 1−3.57T+73T2 1 - 3.57T + 73T^{2}
79 1−10.1T+79T2 1 - 10.1T + 79T^{2}
83 1−5.53T+83T2 1 - 5.53T + 83T^{2}
89 1−18.3T+89T2 1 - 18.3T + 89T^{2}
97 1−9.09T+97T2 1 - 9.09T + 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

−8.188595454983906867891333338951, −7.74488530850097262552236539979, −6.59722175655101401151623082079, −6.17841452944349235645423143508, −5.33905891410561115365331215249, −5.05195477351417119930915873180, −3.79752454289482616995357452928, −3.12477796909336906577067401104, −1.96996952547322643964347186666, −1.01806165169176687269283320691, 1.01806165169176687269283320691, 1.96996952547322643964347186666, 3.12477796909336906577067401104, 3.79752454289482616995357452928, 5.05195477351417119930915873180, 5.33905891410561115365331215249, 6.17841452944349235645423143508, 6.59722175655101401151623082079, 7.74488530850097262552236539979, 8.188595454983906867891333338951

Graph of the ZZ-function along the critical line