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Properties

Label 2-4002-1.1-c1-0-5
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 − 0.0471·5-s − 6-s − 3.98·7-s + 8-s + 9-s − 0.0471·10-s − 5.70·11-s − 12-s − 2.39·13-s − 3.98·14-s + 0.0471·15-s + 16-s + 1.20·17-s + 18-s − 3.98·19-s − 0.0471·20-s + 3.98·21-s − 5.70·22-s + 23-s − 24-s − 4.99·25-s − 2.39·26-s − 27-s − 3.98·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.0210·5-s − 0.408·6-s − 1.50·7-s + 0.353·8-s + 0.333·9-s − 0.0149·10-s − 1.71·11-s − 0.288·12-s − 0.665·13-s − 1.06·14-s + 0.0121·15-s + 0.250·16-s + 0.291·17-s + 0.235·18-s − 0.913·19-s − 0.0105·20-s + 0.869·21-s − 1.21·22-s + 0.208·23-s − 0.204·24-s − 0.999·25-s − 0.470·26-s − 0.192·27-s − 0.752·28-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 1.2643132111.264313211
L(12)L(\frac12) ≈\approx 1.2643132111.264313211
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+0.0471T+5T2 1 + 0.0471T + 5T^{2}
7 1+3.98T+7T2 1 + 3.98T + 7T^{2}
11 1+5.70T+11T2 1 + 5.70T + 11T^{2}
13 1+2.39T+13T2 1 + 2.39T + 13T^{2}
17 1−1.20T+17T2 1 - 1.20T + 17T^{2}
19 1+3.98T+19T2 1 + 3.98T + 19T^{2}
31 1−7.22T+31T2 1 - 7.22T + 31T^{2}
37 1−6.50T+37T2 1 - 6.50T + 37T^{2}
41 1−7.53T+41T2 1 - 7.53T + 41T^{2}
43 1−9.23T+43T2 1 - 9.23T + 43T^{2}
47 1−5.96T+47T2 1 - 5.96T + 47T^{2}
53 1−5.81T+53T2 1 - 5.81T + 53T^{2}
59 1−4.24T+59T2 1 - 4.24T + 59T^{2}
61 1−0.600T+61T2 1 - 0.600T + 61T^{2}
67 1+8.33T+67T2 1 + 8.33T + 67T^{2}
71 1−1.23T+71T2 1 - 1.23T + 71T^{2}
73 1−5.54T+73T2 1 - 5.54T + 73T^{2}
79 1+11.1T+79T2 1 + 11.1T + 79T^{2}
83 1−3.18T+83T2 1 - 3.18T + 83T^{2}
89 1+11.3T+89T2 1 + 11.3T + 89T^{2}
97 1+8.49T+97T2 1 + 8.49T + 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

−8.237961655990630483006800877139, −7.50051970224511525974045979271, −6.88243932304139205744709405718, −5.90497917905464754934484115952, −5.74907081881693108670863331065, −4.65077518510608800618745056330, −4.00515728557549110617779926585, −2.81360199195475935709795310286, −2.44963580199349348031386369385, −0.55676331421238503860253861745, 0.55676331421238503860253861745, 2.44963580199349348031386369385, 2.81360199195475935709795310286, 4.00515728557549110617779926585, 4.65077518510608800618745056330, 5.74907081881693108670863331065, 5.90497917905464754934484115952, 6.88243932304139205744709405718, 7.50051970224511525974045979271, 8.237961655990630483006800877139

Graph of the ZZ-function along the critical line