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Properties

Label 2-4002-1.1-c1-0-35
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 + 1.38·5-s + 6-s − 3.20·7-s + 8-s + 9-s + 1.38·10-s + 5.82·11-s + 12-s − 0.957·13-s − 3.20·14-s + 1.38·15-s + 16-s − 3.20·17-s + 18-s + 3.20·19-s + 1.38·20-s − 3.20·21-s + 5.82·22-s − 23-s + 24-s − 3.08·25-s − 0.957·26-s + 27-s − 3.20·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.619·5-s + 0.408·6-s − 1.21·7-s + 0.353·8-s + 0.333·9-s + 0.437·10-s + 1.75·11-s + 0.288·12-s − 0.265·13-s − 0.856·14-s + 0.357·15-s + 0.250·16-s − 0.777·17-s + 0.235·18-s + 0.735·19-s + 0.309·20-s − 0.699·21-s + 1.24·22-s − 0.208·23-s + 0.204·24-s − 0.616·25-s − 0.187·26-s + 0.192·27-s − 0.605·28-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 4.2084655724.208465572
L(12)L(\frac12) ≈\approx 4.2084655724.208465572
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 1−1.38T+5T2 1 - 1.38T + 5T^{2}
7 1+3.20T+7T2 1 + 3.20T + 7T^{2}
11 1−5.82T+11T2 1 - 5.82T + 11T^{2}
13 1+0.957T+13T2 1 + 0.957T + 13T^{2}
17 1+3.20T+17T2 1 + 3.20T + 17T^{2}
19 1−3.20T+19T2 1 - 3.20T + 19T^{2}
31 1−5.72T+31T2 1 - 5.72T + 31T^{2}
37 1−5.22T+37T2 1 - 5.22T + 37T^{2}
41 1+0.0838T+41T2 1 + 0.0838T + 41T^{2}
43 1−9.91T+43T2 1 - 9.91T + 43T^{2}
47 1−8.60T+47T2 1 - 8.60T + 47T^{2}
53 1−12.9T+53T2 1 - 12.9T + 53T^{2}
59 1−1.55T+59T2 1 - 1.55T + 59T^{2}
61 1+11.1T+61T2 1 + 11.1T + 61T^{2}
67 1−8.39T+67T2 1 - 8.39T + 67T^{2}
71 1+11.0T+71T2 1 + 11.0T + 71T^{2}
73 1−9.74T+73T2 1 - 9.74T + 73T^{2}
79 1+8.33T+79T2 1 + 8.33T + 79T^{2}
83 1+10.5T+83T2 1 + 10.5T + 83T^{2}
89 1+8.98T+89T2 1 + 8.98T + 89T^{2}
97 1−7.71T+97T2 1 - 7.71T + 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

−8.607250613063000775493068800738, −7.46228480782137882280889406906, −6.83428656669093386248928786111, −6.21350795132052184310064649521, −5.68078310836492675432732822605, −4.35569426831636427665935057582, −3.93599366292903357364614241732, −2.97527158551406448730150668195, −2.26911511414332033860136081135, −1.08785004485030173178600087229, 1.08785004485030173178600087229, 2.26911511414332033860136081135, 2.97527158551406448730150668195, 3.93599366292903357364614241732, 4.35569426831636427665935057582, 5.68078310836492675432732822605, 6.21350795132052184310064649521, 6.83428656669093386248928786111, 7.46228480782137882280889406906, 8.607250613063000775493068800738

Graph of the ZZ-function along the critical line