↑

Properties

Label 2-190-5.4-c1-0-6
Degree 22
Conductor 190190
Sign 0.948+0.316i0.948 + 0.316i
Analytic cond. 1.517151.51715
Root an. cond. 1.231721.23172
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  + i·2-s − 0.414i·3-s − 4-s + (0.707 − 2.12i)5-s + 0.414·6-s − 4.41i·7-s − i·8-s + 2.82·9-s + (2.12 + 0.707i)10-s − 1.41·11-s + 0.414i·12-s + 5.82i·13-s + 4.41·14-s + (−0.878 − 0.292i)15-s + 16-s − i·17-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.239i·3-s − 0.5·4-s + (0.316 − 0.948i)5-s + 0.169·6-s − 1.66i·7-s − 0.353i·8-s + 0.942·9-s + (0.670 + 0.223i)10-s − 0.426·11-s + 0.119i·12-s + 1.61i·13-s + 1.17·14-s + (−0.226 − 0.0756i)15-s + 0.250·16-s − 0.242i·17-s + ⋯

Functional equation

Λ(s)=(190s/2 ΓC(s) L(s)=((0.948+0.316i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 190 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.948 + 0.316i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(190s/2 ΓC(s+1/2) L(s)=((0.948+0.316i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 190 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.948 + 0.316i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 190190    =    2⋅5⋅192 \cdot 5 \cdot 19
Sign: 0.948+0.316i0.948 + 0.316i
Analytic conductor: 1.517151.51715
Root analytic conductor: 1.231721.23172
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ190(39,⋅)\chi_{190} (39, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 190, ( :1/2), 0.948+0.316i)(2,\ 190,\ (\ :1/2),\ 0.948 + 0.316i)

Particular Values

L(1)L(1) ≈\approx 1.17725−0.191042i1.17725 - 0.191042i
L(12)L(\frac12) ≈\approx 1.17725−0.191042i1.17725 - 0.191042i
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−iT 1 - iT
5 1+(−0.707+2.12i)T 1 + (-0.707 + 2.12i)T
19 1−T 1 - T
good3 1+0.414iT−3T2 1 + 0.414iT - 3T^{2}
7 1+4.41iT−7T2 1 + 4.41iT - 7T^{2}
11 1+1.41T+11T2 1 + 1.41T + 11T^{2}
13 1−5.82iT−13T2 1 - 5.82iT - 13T^{2}
17 1+iT−17T2 1 + iT - 17T^{2}
23 1−0.757iT−23T2 1 - 0.757iT - 23T^{2}
29 1−0.171T+29T2 1 - 0.171T + 29T^{2}
31 1−6.24T+31T2 1 - 6.24T + 31T^{2}
37 1−8.48iT−37T2 1 - 8.48iT - 37T^{2}
41 1+4.24T+41T2 1 + 4.24T + 41T^{2}
43 1−1.75iT−43T2 1 - 1.75iT - 43T^{2}
47 1−47T2 1 - 47T^{2}
53 1−5.48iT−53T2 1 - 5.48iT - 53T^{2}
59 1+6.89T+59T2 1 + 6.89T + 59T^{2}
61 1−14.2T+61T2 1 - 14.2T + 61T^{2}
67 1+4.75iT−67T2 1 + 4.75iT - 67T^{2}
71 1+13.4T+71T2 1 + 13.4T + 71T^{2}
73 1−11.4iT−73T2 1 - 11.4iT - 73T^{2}
79 1−6.48T+79T2 1 - 6.48T + 79T^{2}
83 1−14.4iT−83T2 1 - 14.4iT - 83T^{2}
89 1+7.07T+89T2 1 + 7.07T + 89T^{2}
97 1−0.343iT−97T2 1 - 0.343iT - 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

−12.84026262976644013734149674382, −11.66709833529382198081184685133, −10.20811269563823693358341426021, −9.555864716288353031490040441524, −8.273651025419585526355966086043, −7.26595641815642898297319851997, −6.51151305988833745420259071204, −4.80663747676699940447118266967, −4.13756085135327951424793360402, −1.27961502630059768149103804813, 2.28095108453774903385387795096, 3.31919267666252988739955505355, 5.12621272769236787494226985625, 6.07929888080104895390591115765, 7.64908703923777749395982372292, 8.808212735528691310948521915042, 9.968751386454530848925602530910, 10.47766264112560788309307808799, 11.61628367479862979123327885029, 12.58310225874120778139327131706

Graph of the ZZ-function along the critical line