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Properties

Label 2-3800-760.189-c0-0-3
Degree 22
Conductor 38003800
Sign 0.447−0.894i0.447 - 0.894i
Analytic cond. 1.896441.89644
Root an. cond. 1.377111.37711
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − i·2-s + 1.53i·3-s − 4-s + 1.53·6-s − 0.347i·7-s + i·8-s − 1.34·9-s − 1.53i·12-s − 1.87i·13-s − 0.347·14-s + 16-s + 1.87i·17-s + 1.34i·18-s − 19-s + 0.532·21-s + ⋯
L(s)  = 1  − i·2-s + 1.53i·3-s − 4-s + 1.53·6-s − 0.347i·7-s + i·8-s − 1.34·9-s − 1.53i·12-s − 1.87i·13-s − 0.347·14-s + 16-s + 1.87i·17-s + 1.34i·18-s − 19-s + 0.532·21-s + ⋯

Functional equation

Λ(s)=(3800s/2 ΓC(s) L(s)=((0.447−0.894i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(3800s/2 ΓC(s) L(s)=((0.447−0.894i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, 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: 1.896441.89644
Root analytic conductor: 1.377111.37711
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ3800(949,⋅)\chi_{3800} (949, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3800, ( :0), 0.447−0.894i)(2,\ 3800,\ (\ :0),\ 0.447 - 0.894i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.94957554220.9495755422
L(12)L(\frac12) ≈\approx 0.94957554220.9495755422
L(1)L(1) 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+iT 1 + iT
5 1 1
19 1+T 1 + T
good3 1−1.53iT−T2 1 - 1.53iT - T^{2}
7 1+0.347iT−T2 1 + 0.347iT - T^{2}
11 1−T2 1 - T^{2}
13 1+1.87iT−T2 1 + 1.87iT - T^{2}
17 1−1.87iT−T2 1 - 1.87iT - T^{2}
23 1−1.53iT−T2 1 - 1.53iT - T^{2}
29 1−1.87T+T2 1 - 1.87T + T^{2}
31 1−T2 1 - T^{2}
37 1−iT−T2 1 - iT - T^{2}
41 1−T2 1 - T^{2}
43 1+T2 1 + T^{2}
47 1−iT−T2 1 - iT - T^{2}
53 1−0.347iT−T2 1 - 0.347iT - T^{2}
59 1+1.53T+T2 1 + 1.53T + T^{2}
61 1−T2 1 - T^{2}
67 1+0.347iT−T2 1 + 0.347iT - T^{2}
71 1−T2 1 - T^{2}
73 1−1.53iT−T2 1 - 1.53iT - T^{2}
79 1−T2 1 - T^{2}
83 1+T2 1 + T^{2}
89 1−T2 1 - T^{2}
97 1+T2 1 + T^{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

−8.954768252258581465507969654458, −8.317592686635888725968014212833, −7.80599719833522009170729227285, −6.21069422207521111440581695158, −5.54215302171513635130467628941, −4.75948594425292033401796425479, −4.08343082045360354246643870477, −3.43340084325485757930990958980, −2.76541946939768003642827771560, −1.29774912106516932850348464564, 0.59391308421665691461222734084, 1.95578864965617538534759869238, 2.78718196518846251748011518083, 4.30327564757625793169181641224, 4.83688218040819580912583476817, 5.96098234393175694228500674720, 6.64698682686890215100202189473, 6.88176350504426104081348863350, 7.58613008060066245343639524592, 8.521596461789546927598301523378

Graph of the ZZ-function along the critical line