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Properties

Label 2-3800-152.37-c0-0-16
Degree 22
Conductor 38003800
Sign 0.382−0.923i0.382 - 0.923i
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  + (0.382 + 0.923i)2-s + 1.84·3-s + (−0.707 + 0.707i)4-s + (0.707 + 1.70i)6-s + (−0.923 − 0.382i)8-s + 2.41·9-s − 1.41i·11-s + (−1.30 + 1.30i)12-s + 0.765·13-s − i·16-s + (0.923 + 2.23i)18-s + i·19-s + (1.30 − 0.541i)22-s + (−1.70 − 0.707i)24-s + (0.292 + 0.707i)26-s + 2.61·27-s + ⋯
L(s)  = 1  + (0.382 + 0.923i)2-s + 1.84·3-s + (−0.707 + 0.707i)4-s + (0.707 + 1.70i)6-s + (−0.923 − 0.382i)8-s + 2.41·9-s − 1.41i·11-s + (−1.30 + 1.30i)12-s + 0.765·13-s − i·16-s + (0.923 + 2.23i)18-s + i·19-s + (1.30 − 0.541i)22-s + (−1.70 − 0.707i)24-s + (0.292 + 0.707i)26-s + 2.61·27-s + ⋯

Functional equation

Λ(s)=(3800s/2 ΓC(s) L(s)=((0.382−0.923i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.382 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(3800s/2 ΓC(s) L(s)=((0.382−0.923i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.382 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 38003800    =    23⋅52⋅192^{3} \cdot 5^{2} \cdot 19
Sign: 0.382−0.923i0.382 - 0.923i
Analytic conductor: 1.896441.89644
Root analytic conductor: 1.377111.37711
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ3800(1101,⋅)\chi_{3800} (1101, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3800, ( :0), 0.382−0.923i)(2,\ 3800,\ (\ :0),\ 0.382 - 0.923i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 2.8308876172.830887617
L(12)L(\frac12) ≈\approx 2.8308876172.830887617
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+(−0.382−0.923i)T 1 + (-0.382 - 0.923i)T
5 1 1
19 1−iT 1 - iT
good3 1−1.84T+T2 1 - 1.84T + T^{2}
7 1+T2 1 + T^{2}
11 1+1.41iT−T2 1 + 1.41iT - T^{2}
13 1−0.765T+T2 1 - 0.765T + T^{2}
17 1+T2 1 + T^{2}
23 1+T2 1 + T^{2}
29 1+T2 1 + T^{2}
31 1−T2 1 - T^{2}
37 1+1.84T+T2 1 + 1.84T + T^{2}
41 1−T2 1 - T^{2}
43 1−T2 1 - T^{2}
47 1+T2 1 + T^{2}
53 1+1.84T+T2 1 + 1.84T + T^{2}
59 1+T2 1 + T^{2}
61 1−1.41iT−T2 1 - 1.41iT - T^{2}
67 1−0.765T+T2 1 - 0.765T + T^{2}
71 1−T2 1 - T^{2}
73 1+T2 1 + T^{2}
79 1−T2 1 - T^{2}
83 1−T2 1 - T^{2}
89 1−T2 1 - T^{2}
97 1+0.765iT−T2 1 + 0.765iT - 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

−8.595911731452791349553838536898, −8.159407116191031498226739870339, −7.58094127778790936452492461395, −6.68394701764801132248834245649, −5.99660213745261177032376074684, −5.02551103126484873641973915024, −3.91965350346931360812277424888, −3.50738307903992242220646157842, −2.85398583604329477404047887834, −1.50711970670807164268213917037, 1.54463113061535235245841511351, 2.11982514932773217352444826942, 3.04453720770457470568375216167, 3.64473176446447483324545418151, 4.48059131645615754939940392084, 5.06255089721723305546171957829, 6.50499874624353533460016946167, 7.17018294709930161992722496362, 8.113637942878325027280875805355, 8.638301129172775102737270931588

Graph of the ZZ-function along the critical line