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Properties

Label 2-4002-1.1-c1-0-45
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 + 2.56·5-s − 6-s + 2.56·7-s − 8-s + 9-s − 2.56·10-s + 5.12·11-s + 12-s + 3.12·13-s − 2.56·14-s + 2.56·15-s + 16-s − 2.56·17-s − 18-s − 6.56·19-s + 2.56·20-s + 2.56·21-s − 5.12·22-s − 23-s − 24-s + 1.56·25-s − 3.12·26-s + 27-s + 2.56·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 0.5·4-s + 1.14·5-s − 0.408·6-s + 0.968·7-s − 0.353·8-s + 0.333·9-s − 0.810·10-s + 1.54·11-s + 0.288·12-s + 0.866·13-s − 0.684·14-s + 0.661·15-s + 0.250·16-s − 0.621·17-s − 0.235·18-s − 1.50·19-s + 0.572·20-s + 0.558·21-s − 1.09·22-s − 0.208·23-s − 0.204·24-s + 0.312·25-s − 0.612·26-s + 0.192·27-s + 0.484·28-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 2.7970661472.797066147
L(12)L(\frac12) ≈\approx 2.7970661472.797066147
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 1−2.56T+5T2 1 - 2.56T + 5T^{2}
7 1−2.56T+7T2 1 - 2.56T + 7T^{2}
11 1−5.12T+11T2 1 - 5.12T + 11T^{2}
13 1−3.12T+13T2 1 - 3.12T + 13T^{2}
17 1+2.56T+17T2 1 + 2.56T + 17T^{2}
19 1+6.56T+19T2 1 + 6.56T + 19T^{2}
31 1+31T2 1 + 31T^{2}
37 1−8.56T+37T2 1 - 8.56T + 37T^{2}
41 1−8.56T+41T2 1 - 8.56T + 41T^{2}
43 1−6.56T+43T2 1 - 6.56T + 43T^{2}
47 1+3.68T+47T2 1 + 3.68T + 47T^{2}
53 1+1.12T+53T2 1 + 1.12T + 53T^{2}
59 1−3.68T+59T2 1 - 3.68T + 59T^{2}
61 1+2T+61T2 1 + 2T + 61T^{2}
67 1−7.12T+67T2 1 - 7.12T + 67T^{2}
71 1−8T+71T2 1 - 8T + 71T^{2}
73 1+12.2T+73T2 1 + 12.2T + 73T^{2}
79 1+4.24T+79T2 1 + 4.24T + 79T^{2}
83 1+3.12T+83T2 1 + 3.12T + 83T^{2}
89 1+7.36T+89T2 1 + 7.36T + 89T^{2}
97 1+8T+97T2 1 + 8T + 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

−8.527123084686993755576046113273, −8.008711947197061390407654179175, −6.97687652327412040303079443253, −6.28519106395986755027539087191, −5.82065446891821920264987102351, −4.47554255424077955242147097606, −3.92348113563401435787925842836, −2.54026772422474172275614036592, −1.84737390275365766567595932958, −1.16555889038917173088536466973, 1.16555889038917173088536466973, 1.84737390275365766567595932958, 2.54026772422474172275614036592, 3.92348113563401435787925842836, 4.47554255424077955242147097606, 5.82065446891821920264987102351, 6.28519106395986755027539087191, 6.97687652327412040303079443253, 8.008711947197061390407654179175, 8.527123084686993755576046113273

Graph of the ZZ-function along the critical line