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Properties

Label 2-4002-1.1-c1-0-32
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 + 3.59·5-s + 6-s + 1.23·7-s − 8-s + 9-s − 3.59·10-s + 0.260·11-s − 12-s + 5.91·13-s − 1.23·14-s − 3.59·15-s + 16-s − 4.05·17-s − 18-s + 6.94·19-s + 3.59·20-s − 1.23·21-s − 0.260·22-s + 23-s + 24-s + 7.94·25-s − 5.91·26-s − 27-s + 1.23·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 0.5·4-s + 1.60·5-s + 0.408·6-s + 0.464·7-s − 0.353·8-s + 0.333·9-s − 1.13·10-s + 0.0784·11-s − 0.288·12-s + 1.64·13-s − 0.328·14-s − 0.929·15-s + 0.250·16-s − 0.984·17-s − 0.235·18-s + 1.59·19-s + 0.804·20-s − 0.268·21-s − 0.0554·22-s + 0.208·23-s + 0.204·24-s + 1.58·25-s − 1.16·26-s − 0.192·27-s + 0.232·28-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 1.9530651971.953065197
L(12)L(\frac12) ≈\approx 1.9530651971.953065197
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 1−3.59T+5T2 1 - 3.59T + 5T^{2}
7 1−1.23T+7T2 1 - 1.23T + 7T^{2}
11 1−0.260T+11T2 1 - 0.260T + 11T^{2}
13 1−5.91T+13T2 1 - 5.91T + 13T^{2}
17 1+4.05T+17T2 1 + 4.05T + 17T^{2}
19 1−6.94T+19T2 1 - 6.94T + 19T^{2}
31 1−1.73T+31T2 1 - 1.73T + 31T^{2}
37 1−1.95T+37T2 1 - 1.95T + 37T^{2}
41 1−1.49T+41T2 1 - 1.49T + 41T^{2}
43 1+2.42T+43T2 1 + 2.42T + 43T^{2}
47 1−2.76T+47T2 1 - 2.76T + 47T^{2}
53 1+2.88T+53T2 1 + 2.88T + 53T^{2}
59 1−0.426T+59T2 1 - 0.426T + 59T^{2}
61 1+5.37T+61T2 1 + 5.37T + 61T^{2}
67 1−15.5T+67T2 1 - 15.5T + 67T^{2}
71 1−12.9T+71T2 1 - 12.9T + 71T^{2}
73 1−1.07T+73T2 1 - 1.07T + 73T^{2}
79 1−5.02T+79T2 1 - 5.02T + 79T^{2}
83 1−14.5T+83T2 1 - 14.5T + 83T^{2}
89 1+10.4T+89T2 1 + 10.4T + 89T^{2}
97 1+11.3T+97T2 1 + 11.3T + 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

−8.546230897692057138533108236243, −7.83330739329757817726815651296, −6.70340990599368591092550559535, −6.39604839453110415759367122178, −5.56173003943525570699708667975, −5.05008325526529854422952873931, −3.79117649733069022940862403101, −2.63980689308710913991800128661, −1.64627094991400125478324745277, −1.02653446190447595663648990950, 1.02653446190447595663648990950, 1.64627094991400125478324745277, 2.63980689308710913991800128661, 3.79117649733069022940862403101, 5.05008325526529854422952873931, 5.56173003943525570699708667975, 6.39604839453110415759367122178, 6.70340990599368591092550559535, 7.83330739329757817726815651296, 8.546230897692057138533108236243

Graph of the ZZ-function along the critical line