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Properties

Label 2-3800-5.4-c1-0-64
Degree 22
Conductor 38003800
Sign −0.447+0.894i-0.447 + 0.894i
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  − 0.656i·3-s − 0.656i·7-s + 2.56·9-s − 0.343·11-s − 1.91i·13-s − 4.48i·17-s + 19-s − 0.431·21-s − 3.56i·23-s − 3.65i·27-s − 7.99·29-s + 5.73·31-s + 0.225i·33-s + 4.16i·37-s − 1.25·39-s + ⋯
L(s)  = 1  − 0.379i·3-s − 0.248i·7-s + 0.856·9-s − 0.103·11-s − 0.530i·13-s − 1.08i·17-s + 0.229·19-s − 0.0940·21-s − 0.744i·23-s − 0.703i·27-s − 1.48·29-s + 1.03·31-s + 0.0392i·33-s + 0.685i·37-s − 0.201·39-s + ⋯

Functional equation

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

Invariants

Degree: 22
Conductor: 38003800    =    23⋅52⋅192^{3} \cdot 5^{2} \cdot 19
Sign: −0.447+0.894i-0.447 + 0.894i
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), −0.447+0.894i)(2,\ 3800,\ (\ :1/2),\ -0.447 + 0.894i)

Particular Values

L(1)L(1) ≈\approx 1.5870211481.587021148
L(12)L(\frac12) ≈\approx 1.5870211481.587021148
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+0.656iT−3T2 1 + 0.656iT - 3T^{2}
7 1+0.656iT−7T2 1 + 0.656iT - 7T^{2}
11 1+0.343T+11T2 1 + 0.343T + 11T^{2}
13 1+1.91iT−13T2 1 + 1.91iT - 13T^{2}
17 1+4.48iT−17T2 1 + 4.48iT - 17T^{2}
23 1+3.56iT−23T2 1 + 3.56iT - 23T^{2}
29 1+7.99T+29T2 1 + 7.99T + 29T^{2}
31 1−5.73T+31T2 1 - 5.73T + 31T^{2}
37 1−4.16iT−37T2 1 - 4.16iT - 37T^{2}
41 1+9.08T+41T2 1 + 9.08T + 41T^{2}
43 1−3.51iT−43T2 1 - 3.51iT - 43T^{2}
47 1+3.40iT−47T2 1 + 3.40iT - 47T^{2}
53 1+1.68iT−53T2 1 + 1.68iT - 53T^{2}
59 1−7.82T+59T2 1 - 7.82T + 59T^{2}
61 1+12.4T+61T2 1 + 12.4T + 61T^{2}
67 1+9.48iT−67T2 1 + 9.48iT - 67T^{2}
71 1+9.96T+71T2 1 + 9.96T + 71T^{2}
73 1−7.53iT−73T2 1 - 7.53iT - 73T^{2}
79 1−14.9T+79T2 1 - 14.9T + 79T^{2}
83 1+10.9iT−83T2 1 + 10.9iT - 83T^{2}
89 1−4.19T+89T2 1 - 4.19T + 89T^{2}
97 1+1.73iT−97T2 1 + 1.73iT - 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

−8.112964885168464244104191690574, −7.47634757340038166537035919684, −6.90396263227852816107272314130, −6.15930253904647669945888421527, −5.16466178950363103819083318308, −4.53496222725466273279326686745, −3.55297171408828834017597842906, −2.63909753318428756730844710575, −1.57440518974004092979092703762, −0.47451669956612729669134553388, 1.34930806915059113116715711415, 2.21941951284794359741144319074, 3.52026849561204917171305787522, 4.05181969328075046486303025632, 4.95004410902713635860150955305, 5.71464057947804517782289088387, 6.52144299507814607279830446923, 7.32592192071415454664632087420, 7.935185370146756678814597324560, 8.922112117753794363410118227625

Graph of the ZZ-function along the critical line