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Properties

Label 2-4002-1.1-c1-0-50
Degree 22
Conductor 40024002
Sign −1-1
Analytic cond. 31.956131.9561
Root an. cond. 5.652975.65297
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 3-s + 4-s − 3·5-s − 6-s − 3·7-s − 8-s + 9-s + 3·10-s + 2·11-s + 12-s + 3·14-s − 3·15-s + 16-s + 3·17-s − 18-s − 19-s − 3·20-s − 3·21-s − 2·22-s + 23-s − 24-s + 4·25-s + 27-s − 3·28-s − 29-s + 3·30-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 1/2·4-s − 1.34·5-s − 0.408·6-s − 1.13·7-s − 0.353·8-s + 1/3·9-s + 0.948·10-s + 0.603·11-s + 0.288·12-s + 0.801·14-s − 0.774·15-s + 1/4·16-s + 0.727·17-s − 0.235·18-s − 0.229·19-s − 0.670·20-s − 0.654·21-s − 0.426·22-s + 0.208·23-s − 0.204·24-s + 4/5·25-s + 0.192·27-s − 0.566·28-s − 0.185·29-s + 0.547·30-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: −1-1
Analytic conductor: 31.956131.9561
Root analytic conductor: 5.652975.65297
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 4002, ( :1/2), −1)(2,\ 4002,\ (\ :1/2),\ -1)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
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+3T+pT2 1 + 3 T + p T^{2}
7 1+3T+pT2 1 + 3 T + p T^{2}
11 1−2T+pT2 1 - 2 T + p T^{2}
13 1+pT2 1 + p T^{2}
17 1−3T+pT2 1 - 3 T + p T^{2}
19 1+T+pT2 1 + T + p T^{2}
31 1−8T+pT2 1 - 8 T + p T^{2}
37 1+5T+pT2 1 + 5 T + p T^{2}
41 1−3T+pT2 1 - 3 T + p T^{2}
43 1−T+pT2 1 - T + p T^{2}
47 1+11T+pT2 1 + 11 T + p T^{2}
53 1+6T+pT2 1 + 6 T + p T^{2}
59 1+5T+pT2 1 + 5 T + p T^{2}
61 1−14T+pT2 1 - 14 T + p T^{2}
67 1−12T+pT2 1 - 12 T + p T^{2}
71 1+pT2 1 + p T^{2}
73 1+6T+pT2 1 + 6 T + p T^{2}
79 1−10T+pT2 1 - 10 T + p T^{2}
83 1+8T+pT2 1 + 8 T + p T^{2}
89 1−10T+pT2 1 - 10 T + p T^{2}
97 1+8T+pT2 1 + 8 T + p T^{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

−8.181287385165343335154846329483, −7.51030725198775318183254953675, −6.77913959791946228009258145967, −6.25910920730510727427596292045, −4.97029829415090071042220572418, −3.88357020461952428072968382897, −3.43432195432228343033354965317, −2.60670009310683293711334821473, −1.18955940393403893990759813368, 0, 1.18955940393403893990759813368, 2.60670009310683293711334821473, 3.43432195432228343033354965317, 3.88357020461952428072968382897, 4.97029829415090071042220572418, 6.25910920730510727427596292045, 6.77913959791946228009258145967, 7.51030725198775318183254953675, 8.181287385165343335154846329483

Graph of the ZZ-function along the critical line