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Properties

Label 2-3800-5.4-c1-0-62
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  + 3.08i·3-s + 4.29i·7-s − 6.51·9-s − 1.21·11-s − 5.08i·13-s − 2.29i·17-s + 19-s − 13.2·21-s − 7.67i·23-s − 10.8i·27-s + 0.489·29-s − 3.74i·33-s − 2i·37-s + 15.6·39-s − 4.16·41-s + ⋯
L(s)  = 1  + 1.78i·3-s + 1.62i·7-s − 2.17·9-s − 0.365·11-s − 1.41i·13-s − 0.557i·17-s + 0.229·19-s − 2.89·21-s − 1.60i·23-s − 2.08i·27-s + 0.0909·29-s − 0.651i·33-s − 0.328i·37-s + 2.51·39-s − 0.650·41-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 0.61234913320.6123491332
L(12)L(\frac12) ≈\approx 0.61234913320.6123491332
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 1−3.08iT−3T2 1 - 3.08iT - 3T^{2}
7 1−4.29iT−7T2 1 - 4.29iT - 7T^{2}
11 1+1.21T+11T2 1 + 1.21T + 11T^{2}
13 1+5.08iT−13T2 1 + 5.08iT - 13T^{2}
17 1+2.29iT−17T2 1 + 2.29iT - 17T^{2}
23 1+7.67iT−23T2 1 + 7.67iT - 23T^{2}
29 1−0.489T+29T2 1 - 0.489T + 29T^{2}
31 1+31T2 1 + 31T^{2}
37 1+2iT−37T2 1 + 2iT - 37T^{2}
41 1+4.16T+41T2 1 + 4.16T + 41T^{2}
43 1+12.9iT−43T2 1 + 12.9iT - 43T^{2}
47 1+5.80iT−47T2 1 + 5.80iT - 47T^{2}
53 1+1.93iT−53T2 1 + 1.93iT - 53T^{2}
59 1−11.0T+59T2 1 - 11.0T + 59T^{2}
61 1+5.38T+61T2 1 + 5.38T + 61T^{2}
67 1−2.48iT−67T2 1 - 2.48iT - 67T^{2}
71 1+11.7T+71T2 1 + 11.7T + 71T^{2}
73 1−8.46iT−73T2 1 - 8.46iT - 73T^{2}
79 1+1.83T+79T2 1 + 1.83T + 79T^{2}
83 1−7.02iT−83T2 1 - 7.02iT - 83T^{2}
89 1+13.7T+89T2 1 + 13.7T + 89T^{2}
97 1+3.57iT−97T2 1 + 3.57iT - 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

−8.590870865685355767489217060915, −8.179836837340462387887825771707, −6.86659583918618434549370964103, −5.61204123343592595059291395406, −5.52510651675324663879447766850, −4.79207056184273708729411307530, −3.81834009142702967632493230365, −2.89920171658089562192851956218, −2.45630065209201146777977592173, −0.18117613672399972289656012796, 1.23088984517142501884850294330, 1.64346976005425093249839407508, 2.89891829240674223251911496352, 3.85676706846165223567941759290, 4.76911716785583111884395203470, 5.91339900931436587634682368140, 6.56008320259746766802898049846, 7.16627588304168852570438866281, 7.63160751573488777814412398505, 8.189768577543337456022691851440

Graph of the ZZ-function along the critical line