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Properties

Label 2-3800-5.4-c1-0-37
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  + 3.29i·3-s − 1.78i·7-s − 7.87·9-s − 5.08·11-s − 1.29i·13-s − 0.213i·17-s + 19-s + 5.89·21-s − 3.72i·23-s − 16.0i·27-s − 0.870·29-s − 16.7i·33-s + 2i·37-s + 4.27·39-s + 8.59·41-s + ⋯
L(s)  = 1  + 1.90i·3-s − 0.675i·7-s − 2.62·9-s − 1.53·11-s − 0.359i·13-s − 0.0517i·17-s + 0.229·19-s + 1.28·21-s − 0.776i·23-s − 3.09i·27-s − 0.161·29-s − 2.91i·33-s + 0.328i·37-s + 0.684·39-s + 1.34·41-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 1.1428981911.142898191
L(12)L(\frac12) ≈\approx 1.1428981911.142898191
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 1−3.29iT−3T2 1 - 3.29iT - 3T^{2}
7 1+1.78iT−7T2 1 + 1.78iT - 7T^{2}
11 1+5.08T+11T2 1 + 5.08T + 11T^{2}
13 1+1.29iT−13T2 1 + 1.29iT - 13T^{2}
17 1+0.213iT−17T2 1 + 0.213iT - 17T^{2}
23 1+3.72iT−23T2 1 + 3.72iT - 23T^{2}
29 1+0.870T+29T2 1 + 0.870T + 29T^{2}
31 1+31T2 1 + 31T^{2}
37 1−2iT−37T2 1 - 2iT - 37T^{2}
41 1−8.59T+41T2 1 - 8.59T + 41T^{2}
43 1+3.67iT−43T2 1 + 3.67iT - 43T^{2}
47 1−4.65iT−47T2 1 - 4.65iT - 47T^{2}
53 1−11.0iT−53T2 1 - 11.0iT - 53T^{2}
59 1−4.70T+59T2 1 - 4.70T + 59T^{2}
61 1−3.51T+61T2 1 - 3.51T + 61T^{2}
67 1+1.12iT−67T2 1 + 1.12iT - 67T^{2}
71 1−8.76T+71T2 1 - 8.76T + 71T^{2}
73 1−6.80iT−73T2 1 - 6.80iT - 73T^{2}
79 1+14.5T+79T2 1 + 14.5T + 79T^{2}
83 1+9.74iT−83T2 1 + 9.74iT - 83T^{2}
89 1−6.76T+89T2 1 - 6.76T + 89T^{2}
97 1+4.16iT−97T2 1 + 4.16iT - 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

−8.645238315829141040651353748430, −8.011016614926993063084647821514, −7.23023390637691402810923139053, −5.93837314938285479795521522476, −5.42774559470322649013999145821, −4.62215930456886154674755318837, −4.12452443222169320759764751278, −3.14327578219914648328075433019, −2.54285430051668785870194570838, −0.43633270999248396691422672889, 0.813819482521378409310606198258, 2.06849824965731914520627372508, 2.47746876956439835558153654184, 3.46891268047027817947593825152, 5.05764035846719257121564050303, 5.62708923985748856482085766401, 6.25877460940886423416976043402, 7.14400245054850604366597799711, 7.62961825898824289645098686927, 8.242831992292813961791070088434

Graph of the ZZ-function along the critical line