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Properties

Label 2-4000-20.19-c0-0-5
Degree 22
Conductor 40004000
Sign 0.707+0.707i0.707 + 0.707i
Analytic cond. 1.996261.99626
Root an. cond. 1.412891.41289
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.61·3-s + 7-s + 1.61·9-s − i·11-s + 1.61i·13-s − i·17-s − 1.61i·19-s − 1.61·21-s − 27-s + 29-s + 0.618i·31-s + 1.61i·33-s − 2.61i·39-s + 41-s − 43-s + ⋯
L(s)  = 1  − 1.61·3-s + 7-s + 1.61·9-s − i·11-s + 1.61i·13-s − i·17-s − 1.61i·19-s − 1.61·21-s − 27-s + 29-s + 0.618i·31-s + 1.61i·33-s − 2.61i·39-s + 41-s − 43-s + ⋯

Functional equation

Λ(s)=(4000s/2 ΓC(s) L(s)=((0.707+0.707i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(4000s/2 ΓC(s) L(s)=((0.707+0.707i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 40004000    =    25⋅532^{5} \cdot 5^{3}
Sign: 0.707+0.707i0.707 + 0.707i
Analytic conductor: 1.996261.99626
Root analytic conductor: 1.412891.41289
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ4000(3999,⋅)\chi_{4000} (3999, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 4000, ( :0), 0.707+0.707i)(2,\ 4000,\ (\ :0),\ 0.707 + 0.707i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.79046789480.7904678948
L(12)L(\frac12) ≈\approx 0.79046789480.7904678948
L(1)L(1) 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+1.61T+T2 1 + 1.61T + T^{2}
7 1−T+T2 1 - T + T^{2}
11 1+iT−T2 1 + iT - T^{2}
13 1−1.61iT−T2 1 - 1.61iT - T^{2}
17 1+iT−T2 1 + iT - T^{2}
19 1+1.61iT−T2 1 + 1.61iT - T^{2}
23 1+T2 1 + T^{2}
29 1−T+T2 1 - T + T^{2}
31 1−0.618iT−T2 1 - 0.618iT - T^{2}
37 1−T2 1 - T^{2}
41 1−T+T2 1 - T + T^{2}
43 1+T+T2 1 + T + T^{2}
47 1+0.618T+T2 1 + 0.618T + T^{2}
53 1+0.618iT−T2 1 + 0.618iT - T^{2}
59 1−0.618iT−T2 1 - 0.618iT - T^{2}
61 1+1.61T+T2 1 + 1.61T + T^{2}
67 1−0.618T+T2 1 - 0.618T + T^{2}
71 1+iT−T2 1 + iT - T^{2}
73 1+0.618iT−T2 1 + 0.618iT - T^{2}
79 1−iT−T2 1 - iT - T^{2}
83 1+T2 1 + T^{2}
89 1+T2 1 + T^{2}
97 1+0.618iT−T2 1 + 0.618iT - 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

−8.654547955860649450697028317008, −7.61049651912846960021963812336, −6.78791915377205901158550345637, −6.43187419429404153206293303974, −5.47923043703536918048730434549, −4.70069323542054278298463675761, −4.54452971770920375037687192289, −3.05996549815422088108326301628, −1.77439454314597711757064741245, −0.68376958510417186764408494987, 1.07904556784875570062588343217, 1.98478286030508818142116350718, 3.48592615637549310945588258206, 4.54882166413855651252357203775, 4.96504976219949857300362653745, 5.90806520187116067001145941268, 6.12559588596155914759705362497, 7.28878903727813446718688598513, 7.894652818344594194177023213455, 8.452427230156941721161850141470

Graph of the ZZ-function along the critical line