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Properties

Label 2-3800-5.4-c1-0-13
Degree 22
Conductor 38003800
Sign −0.894−0.447i-0.894 - 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  + 0.642i·3-s + 3.58i·7-s + 2.58·9-s − 1.35i·13-s + 5.58i·17-s − 19-s − 2.30·21-s + 4.87i·23-s + 3.58i·27-s − 9.58·29-s − 7.17·31-s − 0.945i·37-s + 0.871·39-s + 10.4·41-s + 2.71i·43-s + ⋯
L(s)  = 1  + 0.370i·3-s + 1.35i·7-s + 0.862·9-s − 0.376i·13-s + 1.35i·17-s − 0.229·19-s − 0.502·21-s + 1.01i·23-s + 0.690i·27-s − 1.78·29-s − 1.28·31-s − 0.155i·37-s + 0.139·39-s + 1.63·41-s + 0.414i·43-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: −0.894−0.447i-0.894 - 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 1.3555800321.355580032
L(12)L(\frac12) ≈\approx 1.3555800321.355580032
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 1−0.642iT−3T2 1 - 0.642iT - 3T^{2}
7 1−3.58iT−7T2 1 - 3.58iT - 7T^{2}
11 1+11T2 1 + 11T^{2}
13 1+1.35iT−13T2 1 + 1.35iT - 13T^{2}
17 1−5.58iT−17T2 1 - 5.58iT - 17T^{2}
23 1−4.87iT−23T2 1 - 4.87iT - 23T^{2}
29 1+9.58T+29T2 1 + 9.58T + 29T^{2}
31 1+7.17T+31T2 1 + 7.17T + 31T^{2}
37 1+0.945iT−37T2 1 + 0.945iT - 37T^{2}
41 1−10.4T+41T2 1 - 10.4T + 41T^{2}
43 1−2.71iT−43T2 1 - 2.71iT - 43T^{2}
47 1+5.89iT−47T2 1 + 5.89iT - 47T^{2}
53 1+9.81iT−53T2 1 + 9.81iT - 53T^{2}
59 1+10.1T+59T2 1 + 10.1T + 59T^{2}
61 1−3.28T+61T2 1 - 3.28T + 61T^{2}
67 1−10.3iT−67T2 1 - 10.3iT - 67T^{2}
71 1−14.3T+71T2 1 - 14.3T + 71T^{2}
73 1−4.15iT−73T2 1 - 4.15iT - 73T^{2}
79 1+1.28T+79T2 1 + 1.28T + 79T^{2}
83 1−11.1iT−83T2 1 - 11.1iT - 83T^{2}
89 1−6.45T+89T2 1 - 6.45T + 89T^{2}
97 1+13.4iT−97T2 1 + 13.4iT - 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

−8.968708136977542248444540908585, −8.086405394290603406167985462812, −7.45420776331670077984164848906, −6.51435156681031873418921353191, −5.59536003204884577479617124482, −5.33584631366324475094662907077, −4.05960019480092712013542013190, −3.56428376748819566317412804000, −2.30883090086229616369517314445, −1.56372697768125446551306778271, 0.39143926442762275536399869671, 1.41903135605133277498661569868, 2.45743926949288468244277081776, 3.69165310339962767369560678964, 4.28796108798481282090230690788, 5.02279608309189650407825623625, 6.15247918805716268122484083626, 6.87942598608891239478802721835, 7.53182315848715269852763828935, 7.74432118609443604928201055271

Graph of the ZZ-function along the critical line