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Properties

Label 2-20e2-5.4-c3-0-22
Degree 22
Conductor 400400
Sign −0.894+0.447i-0.894 + 0.447i
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  − 7i·3-s + 6i·7-s − 22·9-s + 43·11-s − 28i·13-s − 91i·17-s − 35·19-s + 42·21-s − 162i·23-s − 35i·27-s − 160·29-s − 42·31-s − 301i·33-s + 314i·37-s − 196·39-s + ⋯
L(s)  = 1  − 1.34i·3-s + 0.323i·7-s − 0.814·9-s + 1.17·11-s − 0.597i·13-s − 1.29i·17-s − 0.422·19-s + 0.436·21-s − 1.46i·23-s − 0.249i·27-s − 1.02·29-s − 0.243·31-s − 1.58i·33-s + 1.39i·37-s − 0.804·39-s + ⋯

Functional equation

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

Invariants

Degree: 22
Conductor: 400400    =    24⋅522^{4} \cdot 5^{2}
Sign: −0.894+0.447i-0.894 + 0.447i
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), −0.894+0.447i)(2,\ 400,\ (\ :3/2),\ -0.894 + 0.447i)

Particular Values

L(2)L(2) ≈\approx 1.5467929501.546792950
L(12)L(\frac12) ≈\approx 1.5467929501.546792950
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+7iT−27T2 1 + 7iT - 27T^{2}
7 1−6iT−343T2 1 - 6iT - 343T^{2}
11 1−43T+1.33e3T2 1 - 43T + 1.33e3T^{2}
13 1+28iT−2.19e3T2 1 + 28iT - 2.19e3T^{2}
17 1+91iT−4.91e3T2 1 + 91iT - 4.91e3T^{2}
19 1+35T+6.85e3T2 1 + 35T + 6.85e3T^{2}
23 1+162iT−1.21e4T2 1 + 162iT - 1.21e4T^{2}
29 1+160T+2.43e4T2 1 + 160T + 2.43e4T^{2}
31 1+42T+2.97e4T2 1 + 42T + 2.97e4T^{2}
37 1−314iT−5.06e4T2 1 - 314iT - 5.06e4T^{2}
41 1+203T+6.89e4T2 1 + 203T + 6.89e4T^{2}
43 1+92iT−7.95e4T2 1 + 92iT - 7.95e4T^{2}
47 1−196iT−1.03e5T2 1 - 196iT - 1.03e5T^{2}
53 1−82iT−1.48e5T2 1 - 82iT - 1.48e5T^{2}
59 1+280T+2.05e5T2 1 + 280T + 2.05e5T^{2}
61 1+518T+2.26e5T2 1 + 518T + 2.26e5T^{2}
67 1−141iT−3.00e5T2 1 - 141iT - 3.00e5T^{2}
71 1+412T+3.57e5T2 1 + 412T + 3.57e5T^{2}
73 1+763iT−3.89e5T2 1 + 763iT - 3.89e5T^{2}
79 1−510T+4.93e5T2 1 - 510T + 4.93e5T^{2}
83 1+777iT−5.71e5T2 1 + 777iT - 5.71e5T^{2}
89 1−945T+7.04e5T2 1 - 945T + 7.04e5T^{2}
97 1+1.24e3iT−9.12e5T2 1 + 1.24e3iT - 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

−10.58124976652088098601141289543, −9.356631225121501439067614207365, −8.524008431174238540375117713795, −7.52331073564011179322902942975, −6.73173794090874217727267818602, −5.96207732822383775842538285837, −4.58604461128653227671891498493, −2.99649552687594516601490132911, −1.77430323532526187344602828642, −0.52430139752224847275078268789, 1.67333006552176801852205846904, 3.72714629973999664474696684080, 4.00718254765487309133302652114, 5.30984016629710464406361477787, 6.38556181494375497412423532020, 7.55290888388414223994073033392, 8.916328029373309850884494876079, 9.367365767009723180605233287437, 10.33878462683401464361002083989, 11.04800686206738586350046521188

Graph of the ZZ-function along the critical line