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Properties

Label 2-1200-5.4-c3-0-19
Degree 22
Conductor 12001200
Sign 0.894−0.447i0.894 - 0.447i
Analytic cond. 70.802270.8022
Root an. cond. 8.414408.41440
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  − 3i·3-s − 4i·7-s − 9·9-s + 48·11-s + 2i·13-s + 114i·17-s + 140·19-s − 12·21-s − 72i·23-s + 27i·27-s − 210·29-s − 272·31-s − 144i·33-s + 334i·37-s + 6·39-s + ⋯
L(s)  = 1  − 0.577i·3-s − 0.215i·7-s − 0.333·9-s + 1.31·11-s + 0.0426i·13-s + 1.62i·17-s + 1.69·19-s − 0.124·21-s − 0.652i·23-s + 0.192i·27-s − 1.34·29-s − 1.57·31-s − 0.759i·33-s + 1.48i·37-s + 0.0246·39-s + ⋯

Functional equation

Λ(s)=(1200s/2 ΓC(s) L(s)=((0.894−0.447i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(1200s/2 ΓC(s+3/2) L(s)=((0.894−0.447i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{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: 12001200    =    24⋅3⋅522^{4} \cdot 3 \cdot 5^{2}
Sign: 0.894−0.447i0.894 - 0.447i
Analytic conductor: 70.802270.8022
Root analytic conductor: 8.414408.41440
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ1200(49,⋅)\chi_{1200} (49, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1200, ( :3/2), 0.894−0.447i)(2,\ 1200,\ (\ :3/2),\ 0.894 - 0.447i)

Particular Values

L(2)L(2) ≈\approx 2.0637333992.063733399
L(12)L(\frac12) ≈\approx 2.0637333992.063733399
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
3 1+3iT 1 + 3iT
5 1 1
good7 1+4iT−343T2 1 + 4iT - 343T^{2}
11 1−48T+1.33e3T2 1 - 48T + 1.33e3T^{2}
13 1−2iT−2.19e3T2 1 - 2iT - 2.19e3T^{2}
17 1−114iT−4.91e3T2 1 - 114iT - 4.91e3T^{2}
19 1−140T+6.85e3T2 1 - 140T + 6.85e3T^{2}
23 1+72iT−1.21e4T2 1 + 72iT - 1.21e4T^{2}
29 1+210T+2.43e4T2 1 + 210T + 2.43e4T^{2}
31 1+272T+2.97e4T2 1 + 272T + 2.97e4T^{2}
37 1−334iT−5.06e4T2 1 - 334iT - 5.06e4T^{2}
41 1+198T+6.89e4T2 1 + 198T + 6.89e4T^{2}
43 1−268iT−7.95e4T2 1 - 268iT - 7.95e4T^{2}
47 1−216iT−1.03e5T2 1 - 216iT - 1.03e5T^{2}
53 1+78iT−1.48e5T2 1 + 78iT - 1.48e5T^{2}
59 1−240T+2.05e5T2 1 - 240T + 2.05e5T^{2}
61 1−302T+2.26e5T2 1 - 302T + 2.26e5T^{2}
67 1−596iT−3.00e5T2 1 - 596iT - 3.00e5T^{2}
71 1−768T+3.57e5T2 1 - 768T + 3.57e5T^{2}
73 1+478iT−3.89e5T2 1 + 478iT - 3.89e5T^{2}
79 1+640T+4.93e5T2 1 + 640T + 4.93e5T^{2}
83 1−348iT−5.71e5T2 1 - 348iT - 5.71e5T^{2}
89 1+210T+7.04e5T2 1 + 210T + 7.04e5T^{2}
97 1−1.53e3iT−9.12e5T2 1 - 1.53e3iT - 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

−9.359799366833261840636254597388, −8.605876664799966058516507915818, −7.72943548797226142481689761762, −6.94910197208657226420546753877, −6.19560886629945639039652644543, −5.34628656762851323706343060284, −4.05721486601111057134018429552, −3.32786197715328077176298711000, −1.83957999922518805068485831117, −1.05292629844596158041362267066, 0.56398853839584598481768331207, 1.94103194222554993707145549004, 3.31210587949333384787610363608, 3.92338759499866711659654893239, 5.23391451551795439624459672109, 5.63641496991596172262758155199, 7.05710802281960569950249482998, 7.44456820201042208779106791273, 8.871559736343740868445159615325, 9.312330133089882012630771800149

Graph of the ZZ-function along the critical line