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Properties

Label 2-4000-5.4-c1-0-71
Degree 22
Conductor 40004000
Sign ii
Analytic cond. 31.940131.9401
Root an. cond. 5.651565.65156
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.17i·3-s − 1.90i·7-s + 1.61·9-s + 3.80·11-s − 1.23i·13-s − 2i·17-s + 6.15·19-s − 2.23·21-s − 1.90i·23-s − 5.42i·27-s − 3.61·29-s − 0.898·31-s − 4.47i·33-s − 9.70i·37-s − 1.45·39-s + ⋯
L(s)  = 1  − 0.678i·3-s − 0.718i·7-s + 0.539·9-s + 1.14·11-s − 0.342i·13-s − 0.485i·17-s + 1.41·19-s − 0.487·21-s − 0.396i·23-s − 1.04i·27-s − 0.671·29-s − 0.161·31-s − 0.778i·33-s − 1.59i·37-s − 0.232·39-s + ⋯

Functional equation

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

Invariants

Degree: 22
Conductor: 40004000    =    25⋅532^{5} \cdot 5^{3}
Sign: ii
Analytic conductor: 31.940131.9401
Root analytic conductor: 5.651565.65156
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ4000(1249,⋅)\chi_{4000} (1249, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 4000, ( :1/2), i)(2,\ 4000,\ (\ :1/2),\ i)

Particular Values

L(1)L(1) ≈\approx 2.2986438762.298643876
L(12)L(\frac12) ≈\approx 2.2986438762.298643876
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+1.17iT−3T2 1 + 1.17iT - 3T^{2}
7 1+1.90iT−7T2 1 + 1.90iT - 7T^{2}
11 1−3.80T+11T2 1 - 3.80T + 11T^{2}
13 1+1.23iT−13T2 1 + 1.23iT - 13T^{2}
17 1+2iT−17T2 1 + 2iT - 17T^{2}
19 1−6.15T+19T2 1 - 6.15T + 19T^{2}
23 1+1.90iT−23T2 1 + 1.90iT - 23T^{2}
29 1+3.61T+29T2 1 + 3.61T + 29T^{2}
31 1+0.898T+31T2 1 + 0.898T + 31T^{2}
37 1+9.70iT−37T2 1 + 9.70iT - 37T^{2}
41 1−0.854T+41T2 1 - 0.854T + 41T^{2}
43 1−11.1iT−43T2 1 - 11.1iT - 43T^{2}
47 1−9.68iT−47T2 1 - 9.68iT - 47T^{2}
53 1−4iT−53T2 1 - 4iT - 53T^{2}
59 1−3.80T+59T2 1 - 3.80T + 59T^{2}
61 1−1.38T+61T2 1 - 1.38T + 61T^{2}
67 1−1.45iT−67T2 1 - 1.45iT - 67T^{2}
71 1−9.06T+71T2 1 - 9.06T + 71T^{2}
73 1−6.76iT−73T2 1 - 6.76iT - 73T^{2}
79 1−12.8T+79T2 1 - 12.8T + 79T^{2}
83 1+4.25iT−83T2 1 + 4.25iT - 83T^{2}
89 1−3.09T+89T2 1 - 3.09T + 89T^{2}
97 1+14.1iT−97T2 1 + 14.1iT - 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

−7.953462630937507952811184385287, −7.50517379016886775508704672642, −6.92082622386750915773764926383, −6.23193320396536503944740675729, −5.34344319264267434810573642829, −4.33539925988078962487446680140, −3.72758492168214222430167203896, −2.65915326158009069955581425347, −1.45298808475920547287780713387, −0.78590897549567538238694366799, 1.21915368975139066175758718452, 2.17421026116636202656925130299, 3.55433104084113324025209976486, 3.83197935541718894506519066525, 4.98722792959635078350446134896, 5.46214376746942617035574346371, 6.49341484909456474117234029969, 7.06598791335246259794714975198, 7.964515557787612046999658105965, 8.881942718008516126645703889744

Graph of the ZZ-function along the critical line