↑

Properties

Label 2-3762-209.208-c1-0-48
Degree 22
Conductor 37623762
Sign 0.356−0.934i0.356 - 0.934i
Analytic cond. 30.039730.0397
Root an. cond. 5.480855.48085
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s + 2.09·5-s + 2.64i·7-s + 8-s + 2.09·10-s + (0.183 + 3.31i)11-s + 5.69·13-s + 2.64i·14-s + 16-s + 6.27i·17-s + (−3.98 − 1.77i)19-s + 2.09·20-s + (0.183 + 3.31i)22-s + 6.58·23-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.5·4-s + 0.935·5-s + 1.00i·7-s + 0.353·8-s + 0.661·10-s + (0.0552 + 0.998i)11-s + 1.58·13-s + 0.707i·14-s + 0.250·16-s + 1.52i·17-s + (−0.913 − 0.407i)19-s + 0.467·20-s + (0.0391 + 0.706i)22-s + 1.37·23-s + ⋯

Functional equation

Λ(s)=(3762s/2 ΓC(s) L(s)=((0.356−0.934i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 3762 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.356 - 0.934i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(3762s/2 ΓC(s+1/2) L(s)=((0.356−0.934i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3762 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.356 - 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 37623762    =    2⋅32⋅11⋅192 \cdot 3^{2} \cdot 11 \cdot 19
Sign: 0.356−0.934i0.356 - 0.934i
Analytic conductor: 30.039730.0397
Root analytic conductor: 5.480855.48085
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ3762(2089,⋅)\chi_{3762} (2089, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3762, ( :1/2), 0.356−0.934i)(2,\ 3762,\ (\ :1/2),\ 0.356 - 0.934i)

Particular Values

L(1)L(1) ≈\approx 3.7673824653.767382465
L(12)L(\frac12) ≈\approx 3.7673824653.767382465
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−T 1 - T
3 1 1
11 1+(−0.183−3.31i)T 1 + (-0.183 - 3.31i)T
19 1+(3.98+1.77i)T 1 + (3.98 + 1.77i)T
good5 1−2.09T+5T2 1 - 2.09T + 5T^{2}
7 1−2.64iT−7T2 1 - 2.64iT - 7T^{2}
13 1−5.69T+13T2 1 - 5.69T + 13T^{2}
17 1−6.27iT−17T2 1 - 6.27iT - 17T^{2}
23 1−6.58T+23T2 1 - 6.58T + 23T^{2}
29 1+4.30T+29T2 1 + 4.30T + 29T^{2}
31 1−2.68iT−31T2 1 - 2.68iT - 31T^{2}
37 1−1.22iT−37T2 1 - 1.22iT - 37T^{2}
41 1+9.90T+41T2 1 + 9.90T + 41T^{2}
43 1+9.42iT−43T2 1 + 9.42iT - 43T^{2}
47 1+11.2T+47T2 1 + 11.2T + 47T^{2}
53 1+3.73iT−53T2 1 + 3.73iT - 53T^{2}
59 1−0.0279iT−59T2 1 - 0.0279iT - 59T^{2}
61 1−8.90iT−61T2 1 - 8.90iT - 61T^{2}
67 1+15.9iT−67T2 1 + 15.9iT - 67T^{2}
71 1+1.33iT−71T2 1 + 1.33iT - 71T^{2}
73 1−2.05iT−73T2 1 - 2.05iT - 73T^{2}
79 1−15.3T+79T2 1 - 15.3T + 79T^{2}
83 1−6.89iT−83T2 1 - 6.89iT - 83T^{2}
89 1−6.24iT−89T2 1 - 6.24iT - 89T^{2}
97 1−9.05iT−97T2 1 - 9.05iT - 97T^{2}
show more
show less
   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

−8.730906898705466758275468389458, −8.005160511604629566839064997142, −6.67508002639865229848256931821, −6.46811093599464403550442361447, −5.58336349210873998850130261679, −5.07886724202649883686105438103, −4.01706262322895071095661483038, −3.23941853728969268645907527189, −2.02790415872803611109030531707, −1.66663747529418024475269957777, 0.842860406477504337689463847613, 1.80551963551099661039233931228, 3.06162045607237118640734693501, 3.62720159660620487911987850256, 4.59131146586347539983566886425, 5.40065405297556161880543167389, 6.15911246684450351973466868272, 6.62540040867755394079449395365, 7.48052206070989617765454869102, 8.365052942977394996046004158458

Graph of the ZZ-function along the critical line