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Properties

Label 2-3800-5.4-c1-0-60
Degree 22
Conductor 38003800
Sign 0.447+0.894i0.447 + 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  − 0.848i·3-s + 1.74i·7-s + 2.28·9-s + 5.92·11-s − 6.78i·13-s + 1.86i·17-s + 19-s + 1.48·21-s − 5.94i·23-s − 4.47i·27-s − 3.29·29-s − 5.75·31-s − 5.02i·33-s − 4.36i·37-s − 5.75·39-s + ⋯
L(s)  = 1  − 0.489i·3-s + 0.659i·7-s + 0.760·9-s + 1.78·11-s − 1.88i·13-s + 0.451i·17-s + 0.229·19-s + 0.322·21-s − 1.23i·23-s − 0.862i·27-s − 0.612·29-s − 1.03·31-s − 0.874i·33-s − 0.717i·37-s − 0.921·39-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+0.894i0.447 + 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 2.2709157422.270915742
L(12)L(\frac12) ≈\approx 2.2709157422.270915742
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+0.848iT−3T2 1 + 0.848iT - 3T^{2}
7 1−1.74iT−7T2 1 - 1.74iT - 7T^{2}
11 1−5.92T+11T2 1 - 5.92T + 11T^{2}
13 1+6.78iT−13T2 1 + 6.78iT - 13T^{2}
17 1−1.86iT−17T2 1 - 1.86iT - 17T^{2}
23 1+5.94iT−23T2 1 + 5.94iT - 23T^{2}
29 1+3.29T+29T2 1 + 3.29T + 29T^{2}
31 1+5.75T+31T2 1 + 5.75T + 31T^{2}
37 1+4.36iT−37T2 1 + 4.36iT - 37T^{2}
41 1−7.12T+41T2 1 - 7.12T + 41T^{2}
43 1−6.98iT−43T2 1 - 6.98iT - 43T^{2}
47 1+4.02iT−47T2 1 + 4.02iT - 47T^{2}
53 1−9.19iT−53T2 1 - 9.19iT - 53T^{2}
59 1+2.51T+59T2 1 + 2.51T + 59T^{2}
61 1+2.49T+61T2 1 + 2.49T + 61T^{2}
67 1−6.90iT−67T2 1 - 6.90iT - 67T^{2}
71 1−1.27T+71T2 1 - 1.27T + 71T^{2}
73 1+12.1iT−73T2 1 + 12.1iT - 73T^{2}
79 1−13.8T+79T2 1 - 13.8T + 79T^{2}
83 1−4.94iT−83T2 1 - 4.94iT - 83T^{2}
89 1+15.6T+89T2 1 + 15.6T + 89T^{2}
97 1+15.7iT−97T2 1 + 15.7iT - 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

−8.324655027614379535576030737199, −7.60869879361256874068967772749, −6.93046094990278175217397234404, −6.08716290866800015680594243099, −5.63440434947250956652456405853, −4.46613191150594795364263193140, −3.74390137480486905519075226980, −2.74812043531637632592192925961, −1.69903707499535586907531091451, −0.75878320751210343692317020396, 1.22105117527398968286102999411, 1.91303258702036406006892527806, 3.60018044465088418457920987096, 3.96796751526747506118423040006, 4.58887128385372017041781616548, 5.60095463038775083830483832420, 6.80304683040596534375394544574, 6.87712206170750253262522843812, 7.74511386516323201435774703777, 9.014812229529090606949252830806

Graph of the ZZ-function along the critical line