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Properties

Label 2-2040-5.4-c1-0-6
Degree 22
Conductor 20402040
Sign 0.447−0.894i0.447 - 0.894i
Analytic cond. 16.289416.2894
Root an. cond. 4.036024.03602
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  + i·3-s + (−2 − i)5-s − 2i·7-s − 9-s − 6·11-s − 2i·13-s + (1 − 2i)15-s + i·17-s + 2·21-s + 4i·23-s + (3 + 4i)25-s − i·27-s + 8·31-s − 6i·33-s + (−2 + 4i)35-s + ⋯
L(s)  = 1  + 0.577i·3-s + (−0.894 − 0.447i)5-s − 0.755i·7-s − 0.333·9-s − 1.80·11-s − 0.554i·13-s + (0.258 − 0.516i)15-s + 0.242i·17-s + 0.436·21-s + 0.834i·23-s + (0.600 + 0.800i)25-s − 0.192i·27-s + 1.43·31-s − 1.04i·33-s + (−0.338 + 0.676i)35-s + ⋯

Functional equation

Λ(s)=(2040s/2 ΓC(s) L(s)=((0.447−0.894i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 2040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(2040s/2 ΓC(s+1/2) L(s)=((0.447−0.894i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 20402040    =    23⋅3⋅5⋅172^{3} \cdot 3 \cdot 5 \cdot 17
Sign: 0.447−0.894i0.447 - 0.894i
Analytic conductor: 16.289416.2894
Root analytic conductor: 4.036024.03602
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ2040(409,⋅)\chi_{2040} (409, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2040, ( :1/2), 0.447−0.894i)(2,\ 2040,\ (\ :1/2),\ 0.447 - 0.894i)

Particular Values

L(1)L(1) ≈\approx 0.90895100690.9089510069
L(12)L(\frac12) ≈\approx 0.90895100690.9089510069
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
3 1−iT 1 - iT
5 1+(2+i)T 1 + (2 + i)T
17 1−iT 1 - iT
good7 1+2iT−7T2 1 + 2iT - 7T^{2}
11 1+6T+11T2 1 + 6T + 11T^{2}
13 1+2iT−13T2 1 + 2iT - 13T^{2}
19 1+19T2 1 + 19T^{2}
23 1−4iT−23T2 1 - 4iT - 23T^{2}
29 1+29T2 1 + 29T^{2}
31 1−8T+31T2 1 - 8T + 31T^{2}
37 1−10iT−37T2 1 - 10iT - 37T^{2}
41 1−6T+41T2 1 - 6T + 41T^{2}
43 1−43T2 1 - 43T^{2}
47 1−47T2 1 - 47T^{2}
53 1−10iT−53T2 1 - 10iT - 53T^{2}
59 1−2T+59T2 1 - 2T + 59T^{2}
61 1−10T+61T2 1 - 10T + 61T^{2}
67 1+4iT−67T2 1 + 4iT - 67T^{2}
71 1+71T2 1 + 71T^{2}
73 1+8iT−73T2 1 + 8iT - 73T^{2}
79 1+8T+79T2 1 + 8T + 79T^{2}
83 1+4iT−83T2 1 + 4iT - 83T^{2}
89 1−2T+89T2 1 - 2T + 89T^{2}
97 1+4iT−97T2 1 + 4iT - 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

−9.289561154883854979375807121057, −8.247169377666781407654251839431, −7.911433378652973613321711861174, −7.18398207753420309363629556555, −5.93697951462753148183368962019, −5.04271101581489130900285070541, −4.48415631837275459601018936001, −3.50902819497367371071837608382, −2.69665413170645044314226481025, −0.873241431082976301405158368076, 0.43175973370645989404736552094, 2.35354780256471794224224286057, 2.76625839346010426031716458555, 4.05562548422323838091386482555, 5.04505667660397636341883823770, 5.84511644137409164978919191189, 6.80558640104224134977295148963, 7.46180145379143892792371058079, 8.218653123071807660987326044218, 8.669146525120559742697353251849

Graph of the ZZ-function along the critical line