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Properties

Label 2-4000-1.1-c1-0-28
Degree 22
Conductor 40004000
Sign 11
Analytic cond. 31.940131.9401
Root an. cond. 5.651565.65156
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  + 0.306·3-s + 2.60·7-s − 2.90·9-s − 2.71·11-s + 0.503·13-s + 5.94·17-s + 5.72·19-s + 0.799·21-s − 1.28·23-s − 1.81·27-s + 2.11·29-s − 3.95·31-s − 0.831·33-s + 0.825·37-s + 0.154·39-s − 4.53·41-s + 5.38·43-s − 5.62·47-s − 0.205·49-s + 1.82·51-s + 10.9·53-s + 1.75·57-s + 13.7·59-s − 7.00·61-s − 7.57·63-s − 2.85·67-s − 0.394·69-s + ⋯
L(s)  = 1  + 0.176·3-s + 0.985·7-s − 0.968·9-s − 0.818·11-s + 0.139·13-s + 1.44·17-s + 1.31·19-s + 0.174·21-s − 0.268·23-s − 0.348·27-s + 0.392·29-s − 0.710·31-s − 0.144·33-s + 0.135·37-s + 0.0246·39-s − 0.707·41-s + 0.821·43-s − 0.819·47-s − 0.0294·49-s + 0.255·51-s + 1.50·53-s + 0.232·57-s + 1.79·59-s − 0.896·61-s − 0.954·63-s − 0.348·67-s − 0.0475·69-s + ⋯

Functional equation

Λ(s)=(4000s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(4000s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 40004000    =    25⋅532^{5} \cdot 5^{3}
Sign: 11
Analytic conductor: 31.940131.9401
Root analytic conductor: 5.651565.65156
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 4000, ( :1/2), 1)(2,\ 4000,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.1315825542.131582554
L(12)L(\frac12) ≈\approx 2.1315825542.131582554
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 1
5 1 1
good3 1−0.306T+3T2 1 - 0.306T + 3T^{2}
7 1−2.60T+7T2 1 - 2.60T + 7T^{2}
11 1+2.71T+11T2 1 + 2.71T + 11T^{2}
13 1−0.503T+13T2 1 - 0.503T + 13T^{2}
17 1−5.94T+17T2 1 - 5.94T + 17T^{2}
19 1−5.72T+19T2 1 - 5.72T + 19T^{2}
23 1+1.28T+23T2 1 + 1.28T + 23T^{2}
29 1−2.11T+29T2 1 - 2.11T + 29T^{2}
31 1+3.95T+31T2 1 + 3.95T + 31T^{2}
37 1−0.825T+37T2 1 - 0.825T + 37T^{2}
41 1+4.53T+41T2 1 + 4.53T + 41T^{2}
43 1−5.38T+43T2 1 - 5.38T + 43T^{2}
47 1+5.62T+47T2 1 + 5.62T + 47T^{2}
53 1−10.9T+53T2 1 - 10.9T + 53T^{2}
59 1−13.7T+59T2 1 - 13.7T + 59T^{2}
61 1+7.00T+61T2 1 + 7.00T + 61T^{2}
67 1+2.85T+67T2 1 + 2.85T + 67T^{2}
71 1−11.3T+71T2 1 - 11.3T + 71T^{2}
73 1+10.1T+73T2 1 + 10.1T + 73T^{2}
79 1−11.1T+79T2 1 - 11.1T + 79T^{2}
83 1+5.83T+83T2 1 + 5.83T + 83T^{2}
89 1−13.7T+89T2 1 - 13.7T + 89T^{2}
97 1+7.02T+97T2 1 + 7.02T + 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

−8.315188795635865697496631188165, −7.80428423455454623657033324018, −7.26532758660719514118034074379, −6.02453108061362074549569151098, −5.37887264519669949928535237885, −4.94464768356629311375536181624, −3.67835338587488315604894675673, −2.99364318037871189991859861643, −2.01528364310554876741517759295, −0.843926610585343151757524194620, 0.843926610585343151757524194620, 2.01528364310554876741517759295, 2.99364318037871189991859861643, 3.67835338587488315604894675673, 4.94464768356629311375536181624, 5.37887264519669949928535237885, 6.02453108061362074549569151098, 7.26532758660719514118034074379, 7.80428423455454623657033324018, 8.315188795635865697496631188165

Graph of the ZZ-function along the critical line