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Properties

Label 2-3800-5.4-c1-0-48
Degree 22
Conductor 38003800
Sign 0.894−0.447i0.894 - 0.447i
Analytic cond. 30.343130.3431
Root an. cond. 5.508465.50846
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.786i·3-s + 2.08i·7-s + 2.38·9-s + 1.29·11-s + 1.21i·13-s − 4.08i·17-s + 19-s − 1.63·21-s − 8.95i·23-s + 4.23i·27-s + 9.38·29-s + 1.02i·33-s + 2i·37-s − 0.954·39-s + 3.57·41-s + ⋯
L(s)  = 1  + 0.454i·3-s + 0.787i·7-s + 0.793·9-s + 0.391·11-s + 0.336i·13-s − 0.990i·17-s + 0.229·19-s − 0.357·21-s − 1.86i·23-s + 0.814i·27-s + 1.74·29-s + 0.177i·33-s + 0.328i·37-s − 0.152·39-s + 0.558·41-s + ⋯

Functional equation

Λ(s)=(3800s/2 ΓC(s) L(s)=((0.894−0.447i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(3800s/2 ΓC(s+1/2) L(s)=((0.894−0.447i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 38003800    =    23⋅52⋅192^{3} \cdot 5^{2} \cdot 19
Sign: 0.894−0.447i0.894 - 0.447i
Analytic conductor: 30.343130.3431
Root analytic conductor: 5.508465.50846
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ3800(3649,⋅)\chi_{3800} (3649, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3800, ( :1/2), 0.894−0.447i)(2,\ 3800,\ (\ :1/2),\ 0.894 - 0.447i)

Particular Values

L(1)L(1) ≈\approx 2.2080149102.208014910
L(12)L(\frac12) ≈\approx 2.2080149102.208014910
L(32)L(\frac{3}{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
19 1−T 1 - T
good3 1−0.786iT−3T2 1 - 0.786iT - 3T^{2}
7 1−2.08iT−7T2 1 - 2.08iT - 7T^{2}
11 1−1.29T+11T2 1 - 1.29T + 11T^{2}
13 1−1.21iT−13T2 1 - 1.21iT - 13T^{2}
17 1+4.08iT−17T2 1 + 4.08iT - 17T^{2}
23 1+8.95iT−23T2 1 + 8.95iT - 23T^{2}
29 1−9.38T+29T2 1 - 9.38T + 29T^{2}
31 1+31T2 1 + 31T^{2}
37 1−2iT−37T2 1 - 2iT - 37T^{2}
41 1−3.57T+41T2 1 - 3.57T + 41T^{2}
43 1−7.72iT−43T2 1 - 7.72iT - 43T^{2}
47 1+9.46iT−47T2 1 + 9.46iT - 47T^{2}
53 1+11.9iT−53T2 1 + 11.9iT - 53T^{2}
59 1−7.21T+59T2 1 - 7.21T + 59T^{2}
61 1−4.87T+61T2 1 - 4.87T + 61T^{2}
67 1+11.3iT−67T2 1 + 11.3iT - 67T^{2}
71 1+9.02T+71T2 1 + 9.02T + 71T^{2}
73 1−5.65iT−73T2 1 - 5.65iT - 73T^{2}
79 1+9.57T+79T2 1 + 9.57T + 79T^{2}
83 1−10.7iT−83T2 1 - 10.7iT - 83T^{2}
89 1+11.0T+89T2 1 + 11.0T + 89T^{2}
97 1−8.59iT−97T2 1 - 8.59iT - 97T^{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

−8.670554858659869132898576959691, −7.950691918058362496913368710552, −6.78945406764239395562763070359, −6.59512088330501239084789334218, −5.39658808189692646976470639208, −4.72914277668505858496706884140, −4.10017520662649047483958188422, −2.98651311181765647836770982558, −2.20096637864972396959167303667, −0.876420074098641096445477902280, 0.958907072719369109721223486395, 1.64361203875561117139629037145, 2.93727563754224441665475772800, 3.94189198185279426399946004820, 4.43371693057417858843882396154, 5.61090900370663890151781576155, 6.25654451205843257710606816012, 7.28912623260240307443371042354, 7.39126292465483577564087881675, 8.359358978996834426948912760163

Graph of the ZZ-function along the critical line