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Properties

Label 2-20e2-4.3-c2-0-5
Degree 22
Conductor 400400
Sign −0.500−0.866i-0.500 - 0.866i
Analytic cond. 10.899210.8992
Root an. cond. 3.301393.30139
Motivic weight 22
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.14i·3-s + 9.06i·7-s + 4.41·9-s − 4.28i·11-s + 9.41·13-s − 18·17-s + 36.2i·19-s − 19.4·21-s − 22.9i·23-s + 28.7i·27-s − 44.8·29-s + 35.2i·31-s + 9.16·33-s − 6.58·37-s + 20.1i·39-s + ⋯
L(s)  = 1  + 0.713i·3-s + 1.29i·7-s + 0.490·9-s − 0.389i·11-s + 0.724·13-s − 1.05·17-s + 1.90i·19-s − 0.924·21-s − 0.996i·23-s + 1.06i·27-s − 1.54·29-s + 1.13i·31-s + 0.277·33-s − 0.177·37-s + 0.516i·39-s + ⋯

Functional equation

Λ(s)=(400s/2 ΓC(s) L(s)=((−0.500−0.866i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.500 - 0.866i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(400s/2 ΓC(s+1) L(s)=((−0.500−0.866i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.500 - 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 400400    =    24⋅522^{4} \cdot 5^{2}
Sign: −0.500−0.866i-0.500 - 0.866i
Analytic conductor: 10.899210.8992
Root analytic conductor: 3.301393.30139
Motivic weight: 22
Rational: no
Arithmetic: yes
Character: χ400(351,⋅)\chi_{400} (351, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 400, ( :1), −0.500−0.866i)(2,\ 400,\ (\ :1),\ -0.500 - 0.866i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 0.759495+1.31548i0.759495 + 1.31548i
L(12)L(\frac12) ≈\approx 0.759495+1.31548i0.759495 + 1.31548i
L(2)L(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 1−2.14iT−9T2 1 - 2.14iT - 9T^{2}
7 1−9.06iT−49T2 1 - 9.06iT - 49T^{2}
11 1+4.28iT−121T2 1 + 4.28iT - 121T^{2}
13 1−9.41T+169T2 1 - 9.41T + 169T^{2}
17 1+18T+289T2 1 + 18T + 289T^{2}
19 1−36.2iT−361T2 1 - 36.2iT - 361T^{2}
23 1+22.9iT−529T2 1 + 22.9iT - 529T^{2}
29 1+44.8T+841T2 1 + 44.8T + 841T^{2}
31 1−35.2iT−961T2 1 - 35.2iT - 961T^{2}
37 1+6.58T+1.36e3T2 1 + 6.58T + 1.36e3T^{2}
41 1−52.2T+1.68e3T2 1 - 52.2T + 1.68e3T^{2}
43 1+28.8iT−1.84e3T2 1 + 28.8iT - 1.84e3T^{2}
47 1−90.1iT−2.20e3T2 1 - 90.1iT - 2.20e3T^{2}
53 1+52.2T+2.80e3T2 1 + 52.2T + 2.80e3T^{2}
59 1+17.1iT−3.48e3T2 1 + 17.1iT - 3.48e3T^{2}
61 1+50.5T+3.72e3T2 1 + 50.5T + 3.72e3T^{2}
67 1+33.1iT−4.48e3T2 1 + 33.1iT - 4.48e3T^{2}
71 1+20.1iT−5.04e3T2 1 + 20.1iT - 5.04e3T^{2}
73 1−91.6T+5.32e3T2 1 - 91.6T + 5.32e3T^{2}
79 1−42.8iT−6.24e3T2 1 - 42.8iT - 6.24e3T^{2}
83 1+22.3iT−6.88e3T2 1 + 22.3iT - 6.88e3T^{2}
89 1−47.6T+7.92e3T2 1 - 47.6T + 7.92e3T^{2}
97 1−160.T+9.40e3T2 1 - 160.T + 9.40e3T^{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

−11.15000149856074500658154624326, −10.54752093613568379568362419059, −9.405106244069913301884222809382, −8.834433228648131250019721566610, −7.84996861427044406594145598653, −6.38604832355003618320412593169, −5.61836130131158733517337100905, −4.43567374065122520394415311829, −3.36740599326773650731463560983, −1.85041083219938140380351979676, 0.67313751608410153172722189414, 2.04259590323300974719653698673, 3.78301534386117197411365360535, 4.69902783616073130992015576810, 6.24950296608100761148437604098, 7.18164334360630482821429264944, 7.57910497345413696887346683788, 8.960453388153512301876502061978, 9.834773817080084592803463004735, 10.98054683495287956491086378383

Graph of the ZZ-function along the critical line