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Properties

Label 2-4002-1.1-c1-0-22
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 − 3.78·5-s − 6-s + 3.57·7-s + 8-s + 9-s − 3.78·10-s + 2.34·11-s − 12-s + 4.66·13-s + 3.57·14-s + 3.78·15-s + 16-s − 1.44·17-s + 18-s − 1.40·19-s − 3.78·20-s − 3.57·21-s + 2.34·22-s − 23-s − 24-s + 9.34·25-s + 4.66·26-s − 27-s + 3.57·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 0.5·4-s − 1.69·5-s − 0.408·6-s + 1.35·7-s + 0.353·8-s + 0.333·9-s − 1.19·10-s + 0.706·11-s − 0.288·12-s + 1.29·13-s + 0.954·14-s + 0.977·15-s + 0.250·16-s − 0.350·17-s + 0.235·18-s − 0.321·19-s − 0.846·20-s − 0.779·21-s + 0.499·22-s − 0.208·23-s − 0.204·24-s + 1.86·25-s + 0.915·26-s − 0.192·27-s + 0.675·28-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 2.3074892792.307489279
L(12)L(\frac12) ≈\approx 2.3074892792.307489279
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+3.78T+5T2 1 + 3.78T + 5T^{2}
7 1−3.57T+7T2 1 - 3.57T + 7T^{2}
11 1−2.34T+11T2 1 - 2.34T + 11T^{2}
13 1−4.66T+13T2 1 - 4.66T + 13T^{2}
17 1+1.44T+17T2 1 + 1.44T + 17T^{2}
19 1+1.40T+19T2 1 + 1.40T + 19T^{2}
31 1−1.30T+31T2 1 - 1.30T + 31T^{2}
37 1−8.01T+37T2 1 - 8.01T + 37T^{2}
41 1+7.66T+41T2 1 + 7.66T + 41T^{2}
43 1+5.48T+43T2 1 + 5.48T + 43T^{2}
47 1+6.80T+47T2 1 + 6.80T + 47T^{2}
53 1−0.554T+53T2 1 - 0.554T + 53T^{2}
59 1−3.88T+59T2 1 - 3.88T + 59T^{2}
61 1+0.342T+61T2 1 + 0.342T + 61T^{2}
67 1−11.2T+67T2 1 - 11.2T + 67T^{2}
71 1−15.1T+71T2 1 - 15.1T + 71T^{2}
73 1−0.445T+73T2 1 - 0.445T + 73T^{2}
79 1+5.56T+79T2 1 + 5.56T + 79T^{2}
83 1−8.84T+83T2 1 - 8.84T + 83T^{2}
89 1−7.99T+89T2 1 - 7.99T + 89T^{2}
97 1+8.60T+97T2 1 + 8.60T + 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

−8.202360613154487162192422962891, −7.80625067621528711616615952810, −6.83201704647473664484900793117, −6.32208835229419961039127583041, −5.22465335773459744901128997490, −4.59013464669652377240529275375, −3.98582485600240340930660886389, −3.40340792365937703475036562351, −1.87910477667840933836565232196, −0.847626420433911442941772139968, 0.847626420433911442941772139968, 1.87910477667840933836565232196, 3.40340792365937703475036562351, 3.98582485600240340930660886389, 4.59013464669652377240529275375, 5.22465335773459744901128997490, 6.32208835229419961039127583041, 6.83201704647473664484900793117, 7.80625067621528711616615952810, 8.202360613154487162192422962891

Graph of the ZZ-function along the critical line