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Properties

Label 2-4002-1.1-c1-0-49
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 + 4.15·5-s − 6-s + 0.461·7-s + 8-s + 9-s + 4.15·10-s + 2.29·11-s − 12-s − 2.16·13-s + 0.461·14-s − 4.15·15-s + 16-s + 3.68·17-s + 18-s + 0.461·19-s + 4.15·20-s − 0.461·21-s + 2.29·22-s + 23-s − 24-s + 12.2·25-s − 2.16·26-s − 27-s + 0.461·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 0.5·4-s + 1.85·5-s − 0.408·6-s + 0.174·7-s + 0.353·8-s + 0.333·9-s + 1.31·10-s + 0.691·11-s − 0.288·12-s − 0.600·13-s + 0.123·14-s − 1.07·15-s + 0.250·16-s + 0.894·17-s + 0.235·18-s + 0.105·19-s + 0.929·20-s − 0.100·21-s + 0.488·22-s + 0.208·23-s − 0.204·24-s + 2.45·25-s − 0.424·26-s − 0.192·27-s + 0.0872·28-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 4.0371344884.037134488
L(12)L(\frac12) ≈\approx 4.0371344884.037134488
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 1−4.15T+5T2 1 - 4.15T + 5T^{2}
7 1−0.461T+7T2 1 - 0.461T + 7T^{2}
11 1−2.29T+11T2 1 - 2.29T + 11T^{2}
13 1+2.16T+13T2 1 + 2.16T + 13T^{2}
17 1−3.68T+17T2 1 - 3.68T + 17T^{2}
19 1−0.461T+19T2 1 - 0.461T + 19T^{2}
31 1+2.61T+31T2 1 + 2.61T + 31T^{2}
37 1+7.35T+37T2 1 + 7.35T + 37T^{2}
41 1−1.98T+41T2 1 - 1.98T + 41T^{2}
43 1+4.73T+43T2 1 + 4.73T + 43T^{2}
47 1−12.4T+47T2 1 - 12.4T + 47T^{2}
53 1+7.91T+53T2 1 + 7.91T + 53T^{2}
59 1−13.9T+59T2 1 - 13.9T + 59T^{2}
61 1−7.56T+61T2 1 - 7.56T + 61T^{2}
67 1+1.07T+67T2 1 + 1.07T + 67T^{2}
71 1+7.96T+71T2 1 + 7.96T + 71T^{2}
73 1−14.6T+73T2 1 - 14.6T + 73T^{2}
79 1+16.1T+79T2 1 + 16.1T + 79T^{2}
83 1−1.22T+83T2 1 - 1.22T + 83T^{2}
89 1−1.14T+89T2 1 - 1.14T + 89T^{2}
97 1+13.6T+97T2 1 + 13.6T + 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

−8.550472489428161534067156541147, −7.35747533725142878019463852775, −6.72816406049138503336846695626, −6.10730896872295301801781009240, −5.35152557775888084287255889485, −5.09107484140501474947857933616, −3.95391306704125882280871903823, −2.88268149853655992603567265294, −1.96223556634589965983488624934, −1.18551957697084490724709988211, 1.18551957697084490724709988211, 1.96223556634589965983488624934, 2.88268149853655992603567265294, 3.95391306704125882280871903823, 5.09107484140501474947857933616, 5.35152557775888084287255889485, 6.10730896872295301801781009240, 6.72816406049138503336846695626, 7.35747533725142878019463852775, 8.550472489428161534067156541147

Graph of the ZZ-function along the critical line