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Properties

Label 2-4002-1.1-c1-0-17
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 − 0.652·5-s − 6-s − 3.89·7-s + 8-s + 9-s − 0.652·10-s + 5.43·11-s − 12-s + 3.14·13-s − 3.89·14-s + 0.652·15-s + 16-s + 4.77·17-s + 18-s − 7.24·19-s − 0.652·20-s + 3.89·21-s + 5.43·22-s − 23-s − 24-s − 4.57·25-s + 3.14·26-s − 27-s − 3.89·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.292·5-s − 0.408·6-s − 1.47·7-s + 0.353·8-s + 0.333·9-s − 0.206·10-s + 1.63·11-s − 0.288·12-s + 0.871·13-s − 1.04·14-s + 0.168·15-s + 0.250·16-s + 1.15·17-s + 0.235·18-s − 1.66·19-s − 0.146·20-s + 0.850·21-s + 1.15·22-s − 0.208·23-s − 0.204·24-s − 0.914·25-s + 0.616·26-s − 0.192·27-s − 0.736·28-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 2.1069264642.106926464
L(12)L(\frac12) ≈\approx 2.1069264642.106926464
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+0.652T+5T2 1 + 0.652T + 5T^{2}
7 1+3.89T+7T2 1 + 3.89T + 7T^{2}
11 1−5.43T+11T2 1 - 5.43T + 11T^{2}
13 1−3.14T+13T2 1 - 3.14T + 13T^{2}
17 1−4.77T+17T2 1 - 4.77T + 17T^{2}
19 1+7.24T+19T2 1 + 7.24T + 19T^{2}
31 1−5.19T+31T2 1 - 5.19T + 31T^{2}
37 1+1.28T+37T2 1 + 1.28T + 37T^{2}
41 1+5.34T+41T2 1 + 5.34T + 41T^{2}
43 1+2.02T+43T2 1 + 2.02T + 43T^{2}
47 1−4.83T+47T2 1 - 4.83T + 47T^{2}
53 1−9.97T+53T2 1 - 9.97T + 53T^{2}
59 1−11.0T+59T2 1 - 11.0T + 59T^{2}
61 1+3.43T+61T2 1 + 3.43T + 61T^{2}
67 1+4.99T+67T2 1 + 4.99T + 67T^{2}
71 1+0.305T+71T2 1 + 0.305T + 71T^{2}
73 1+4.98T+73T2 1 + 4.98T + 73T^{2}
79 1−3.65T+79T2 1 - 3.65T + 79T^{2}
83 1−8.46T+83T2 1 - 8.46T + 83T^{2}
89 1−7.44T+89T2 1 - 7.44T + 89T^{2}
97 1−16.6T+97T2 1 - 16.6T + 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

−8.514145209929140007297474725741, −7.43058332928863645183539403113, −6.61725892538844719689741717267, −6.24230625122690068411531527643, −5.74479431948903609799879438951, −4.50075422738214119321475005849, −3.75860635541933623105623822082, −3.41908874728328934385418670915, −2.01520868188976670209533314902, −0.77546503125119835742745990205, 0.77546503125119835742745990205, 2.01520868188976670209533314902, 3.41908874728328934385418670915, 3.75860635541933623105623822082, 4.50075422738214119321475005849, 5.74479431948903609799879438951, 6.24230625122690068411531527643, 6.61725892538844719689741717267, 7.43058332928863645183539403113, 8.514145209929140007297474725741

Graph of the ZZ-function along the critical line