↑

Properties

Label 2-165-5.4-c1-0-10
Degree 22
Conductor 165165
Sign −0.749+0.662i-0.749 + 0.662i
Analytic cond. 1.317531.31753
Root an. cond. 1.147831.14783
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  − 1.48i·2-s − i·3-s − 0.193·4-s + (−1.48 − 1.67i)5-s − 1.48·6-s + 1.19i·7-s − 2.67i·8-s − 9-s + (−2.48 + 2.19i)10-s + 11-s + 0.193i·12-s − 0.806i·13-s + 1.76·14-s + (−1.67 + 1.48i)15-s − 4.35·16-s + 3.76i·17-s + ⋯
L(s)  = 1  − 1.04i·2-s − 0.577i·3-s − 0.0969·4-s + (−0.662 − 0.749i)5-s − 0.604·6-s + 0.451i·7-s − 0.945i·8-s − 0.333·9-s + (−0.784 + 0.693i)10-s + 0.301·11-s + 0.0559i·12-s − 0.223i·13-s + 0.472·14-s + (−0.432 + 0.382i)15-s − 1.08·16-s + 0.913i·17-s + ⋯

Functional equation

Λ(s)=(165s/2 ΓC(s) L(s)=((−0.749+0.662i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.749 + 0.662i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(165s/2 ΓC(s+1/2) L(s)=((−0.749+0.662i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.749 + 0.662i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 165165    =    3⋅5⋅113 \cdot 5 \cdot 11
Sign: −0.749+0.662i-0.749 + 0.662i
Analytic conductor: 1.317531.31753
Root analytic conductor: 1.147831.14783
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ165(34,⋅)\chi_{165} (34, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 165, ( :1/2), −0.749+0.662i)(2,\ 165,\ (\ :1/2),\ -0.749 + 0.662i)

Particular Values

L(1)L(1) ≈\approx 0.395829−1.04521i0.395829 - 1.04521i
L(12)L(\frac12) ≈\approx 0.395829−1.04521i0.395829 - 1.04521i
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)
bad3 1+iT 1 + iT
5 1+(1.48+1.67i)T 1 + (1.48 + 1.67i)T
11 1−T 1 - T
good2 1+1.48iT−2T2 1 + 1.48iT - 2T^{2}
7 1−1.19iT−7T2 1 - 1.19iT - 7T^{2}
13 1+0.806iT−13T2 1 + 0.806iT - 13T^{2}
17 1−3.76iT−17T2 1 - 3.76iT - 17T^{2}
19 1−5.35T+19T2 1 - 5.35T + 19T^{2}
23 1+4iT−23T2 1 + 4iT - 23T^{2}
29 1−4.31T+29T2 1 - 4.31T + 29T^{2}
31 1−0.962T+31T2 1 - 0.962T + 31T^{2}
37 1−1.61iT−37T2 1 - 1.61iT - 37T^{2}
41 1−9.08T+41T2 1 - 9.08T + 41T^{2}
43 1+4.41iT−43T2 1 + 4.41iT - 43T^{2}
47 1−12.3iT−47T2 1 - 12.3iT - 47T^{2}
53 1+1.42iT−53T2 1 + 1.42iT - 53T^{2}
59 1+13.2T+59T2 1 + 13.2T + 59T^{2}
61 1+0.0752T+61T2 1 + 0.0752T + 61T^{2}
67 1+2.70iT−67T2 1 + 2.70iT - 67T^{2}
71 1+14.0T+71T2 1 + 14.0T + 71T^{2}
73 1−10.7iT−73T2 1 - 10.7iT - 73T^{2}
79 1+13.9T+79T2 1 + 13.9T + 79T^{2}
83 1−9.89iT−83T2 1 - 9.89iT - 83T^{2}
89 1+16.8T+89T2 1 + 16.8T + 89T^{2}
97 1−11.4iT−97T2 1 - 11.4iT - 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

−12.35959905347996465801243796378, −11.68108486014377555521997134743, −10.73655288829009720883736592240, −9.476260888961579847257088249322, −8.442288094094856898763813947054, −7.32168950992682340257386495713, −5.95425153960378698893669772119, −4.31851509368664661542632254062, −2.90113401681275246849643611099, −1.21956741581243862017984637287, 3.05639885360294619392388421297, 4.52857750228403223662545657098, 5.86428820859869945912781221266, 7.09756438691985000544857407153, 7.66272035950727495763118504218, 8.987299273180007689724312625251, 10.19906073915398753106600901108, 11.31121787763888967858268662191, 11.86366906208083451104419412544, 13.78953006119763841521639059891

Graph of the ZZ-function along the critical line