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Properties

Label 2-1200-1.1-c3-0-19
Degree 22
Conductor 12001200
Sign 11
Analytic cond. 70.802270.8022
Root an. cond. 8.414408.41440
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·3-s + 16.2·7-s + 9·9-s + 40.2·11-s + 19.7·13-s + 83.0·17-s + 48.8·19-s − 48.6·21-s − 1.61·23-s − 27·27-s − 24.5·29-s + 12.4·31-s − 120.·33-s + 325.·37-s − 59.3·39-s − 242.·41-s − 367.·43-s + 204.·47-s − 80.2·49-s − 249.·51-s − 61.5·53-s − 146.·57-s + 112.·59-s + 477.·61-s + 145.·63-s − 558.·67-s + 4.83·69-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.875·7-s + 0.333·9-s + 1.10·11-s + 0.422·13-s + 1.18·17-s + 0.589·19-s − 0.505·21-s − 0.0146·23-s − 0.192·27-s − 0.157·29-s + 0.0719·31-s − 0.636·33-s + 1.44·37-s − 0.243·39-s − 0.923·41-s − 1.30·43-s + 0.634·47-s − 0.233·49-s − 0.683·51-s − 0.159·53-s − 0.340·57-s + 0.247·59-s + 1.00·61-s + 0.291·63-s − 1.01·67-s + 0.00844·69-s + ⋯

Functional equation

Λ(s)=(1200s/2 ΓC(s) L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(1200s/2 ΓC(s+3/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 12001200    =    24⋅3⋅522^{4} \cdot 3 \cdot 5^{2}
Sign: 11
Analytic conductor: 70.802270.8022
Root analytic conductor: 8.414408.41440
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 1200, ( :3/2), 1)(2,\ 1200,\ (\ :3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 2.3691899292.369189929
L(12)L(\frac12) ≈\approx 2.3691899292.369189929
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+3T 1 + 3T
5 1 1
good7 1−16.2T+343T2 1 - 16.2T + 343T^{2}
11 1−40.2T+1.33e3T2 1 - 40.2T + 1.33e3T^{2}
13 1−19.7T+2.19e3T2 1 - 19.7T + 2.19e3T^{2}
17 1−83.0T+4.91e3T2 1 - 83.0T + 4.91e3T^{2}
19 1−48.8T+6.85e3T2 1 - 48.8T + 6.85e3T^{2}
23 1+1.61T+1.21e4T2 1 + 1.61T + 1.21e4T^{2}
29 1+24.5T+2.43e4T2 1 + 24.5T + 2.43e4T^{2}
31 1−12.4T+2.97e4T2 1 - 12.4T + 2.97e4T^{2}
37 1−325.T+5.06e4T2 1 - 325.T + 5.06e4T^{2}
41 1+242.T+6.89e4T2 1 + 242.T + 6.89e4T^{2}
43 1+367.T+7.95e4T2 1 + 367.T + 7.95e4T^{2}
47 1−204.T+1.03e5T2 1 - 204.T + 1.03e5T^{2}
53 1+61.5T+1.48e5T2 1 + 61.5T + 1.48e5T^{2}
59 1−112.T+2.05e5T2 1 - 112.T + 2.05e5T^{2}
61 1−477.T+2.26e5T2 1 - 477.T + 2.26e5T^{2}
67 1+558.T+3.00e5T2 1 + 558.T + 3.00e5T^{2}
71 1+558.T+3.57e5T2 1 + 558.T + 3.57e5T^{2}
73 1−1.01e3T+3.89e5T2 1 - 1.01e3T + 3.89e5T^{2}
79 1+1.15e3T+4.93e5T2 1 + 1.15e3T + 4.93e5T^{2}
83 1−1.15e3T+5.71e5T2 1 - 1.15e3T + 5.71e5T^{2}
89 1−96.9T+7.04e5T2 1 - 96.9T + 7.04e5T^{2}
97 1+1.15e3T+9.12e5T2 1 + 1.15e3T + 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

−9.478131657047385873086771160621, −8.489527145532970048176229211555, −7.73911056024936449867241792804, −6.84372110389886155498715598437, −5.95799084770097863095443770661, −5.17382705644969217853794916366, −4.25407457691096052783135073048, −3.28569540606277144488958352111, −1.69619500963758275343443964988, −0.892085465030633368401629993636, 0.892085465030633368401629993636, 1.69619500963758275343443964988, 3.28569540606277144488958352111, 4.25407457691096052783135073048, 5.17382705644969217853794916366, 5.95799084770097863095443770661, 6.84372110389886155498715598437, 7.73911056024936449867241792804, 8.489527145532970048176229211555, 9.478131657047385873086771160621

Graph of the ZZ-function along the critical line