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Properties

Label 2-20e2-5.4-c3-0-18
Degree 22
Conductor 400400
Sign −0.447+0.894i-0.447 + 0.894i
Analytic cond. 23.600723.6007
Root an. cond. 4.858064.85806
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 6i·3-s + 34i·7-s − 9·9-s − 16·11-s − 58i·13-s − 70i·17-s + 4·19-s + 204·21-s − 134i·23-s − 108i·27-s + 242·29-s − 100·31-s + 96i·33-s − 438i·37-s − 348·39-s + ⋯
L(s)  = 1  − 1.15i·3-s + 1.83i·7-s − 0.333·9-s − 0.438·11-s − 1.23i·13-s − 0.998i·17-s + 0.0482·19-s + 2.11·21-s − 1.21i·23-s − 0.769i·27-s + 1.54·29-s − 0.579·31-s + 0.506i·33-s − 1.94i·37-s − 1.42·39-s + ⋯

Functional equation

Λ(s)=(400s/2 ΓC(s) L(s)=((−0.447+0.894i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(400s/2 ΓC(s+3/2) L(s)=((−0.447+0.894i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 400400    =    24⋅522^{4} \cdot 5^{2}
Sign: −0.447+0.894i-0.447 + 0.894i
Analytic conductor: 23.600723.6007
Root analytic conductor: 4.858064.85806
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ400(49,⋅)\chi_{400} (49, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 400, ( :3/2), −0.447+0.894i)(2,\ 400,\ (\ :3/2),\ -0.447 + 0.894i)

Particular Values

L(2)L(2) ≈\approx 1.4739994431.473999443
L(12)L(\frac12) ≈\approx 1.4739994431.473999443
L(52)L(\frac{5}{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+6iT−27T2 1 + 6iT - 27T^{2}
7 1−34iT−343T2 1 - 34iT - 343T^{2}
11 1+16T+1.33e3T2 1 + 16T + 1.33e3T^{2}
13 1+58iT−2.19e3T2 1 + 58iT - 2.19e3T^{2}
17 1+70iT−4.91e3T2 1 + 70iT - 4.91e3T^{2}
19 1−4T+6.85e3T2 1 - 4T + 6.85e3T^{2}
23 1+134iT−1.21e4T2 1 + 134iT - 1.21e4T^{2}
29 1−242T+2.43e4T2 1 - 242T + 2.43e4T^{2}
31 1+100T+2.97e4T2 1 + 100T + 2.97e4T^{2}
37 1+438iT−5.06e4T2 1 + 438iT - 5.06e4T^{2}
41 1+138T+6.89e4T2 1 + 138T + 6.89e4T^{2}
43 1−178iT−7.95e4T2 1 - 178iT - 7.95e4T^{2}
47 1+22iT−1.03e5T2 1 + 22iT - 1.03e5T^{2}
53 1+162iT−1.48e5T2 1 + 162iT - 1.48e5T^{2}
59 1+268T+2.05e5T2 1 + 268T + 2.05e5T^{2}
61 1−250T+2.26e5T2 1 - 250T + 2.26e5T^{2}
67 1+422iT−3.00e5T2 1 + 422iT - 3.00e5T^{2}
71 1−852T+3.57e5T2 1 - 852T + 3.57e5T^{2}
73 1+306iT−3.89e5T2 1 + 306iT - 3.89e5T^{2}
79 1+456T+4.93e5T2 1 + 456T + 4.93e5T^{2}
83 1−434iT−5.71e5T2 1 - 434iT - 5.71e5T^{2}
89 1−726T+7.04e5T2 1 - 726T + 7.04e5T^{2}
97 1−1.37e3iT−9.12e5T2 1 - 1.37e3iT - 9.12e5T^{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

−10.65826032989368437312712902917, −9.497745797918961760249217369403, −8.501489940770908291551762169589, −7.86478149262175088128876572911, −6.74282190826172659511437740234, −5.82456971045870520194256828519, −4.98514548673724351049126967314, −2.90170012563987265123658771305, −2.18412211548998227946560321303, −0.51723925781288068149057793760, 1.37921044625640459165007256554, 3.45748659073392774035359063936, 4.19442873377851438843916687330, 4.95670295008927289785875435569, 6.51582678129768632772835033799, 7.37929025720895743040115393577, 8.477831212631703126823508346588, 9.664062918747706105855953769166, 10.23063070210092321225062204636, 10.84647483471407206633882960493

Graph of the ZZ-function along the critical line