Properties

Label 2-1320-11.4-c1-0-8
Degree 22
Conductor 13201320
Sign 0.9700.242i0.970 - 0.242i
Analytic cond. 10.540210.5402
Root an. cond. 3.246573.24657
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  + (0.309 − 0.951i)3-s + (0.809 − 0.587i)5-s + (0.236 + 0.726i)7-s + (−0.809 − 0.587i)9-s + (3.04 + 1.31i)11-s + (4.73 + 3.44i)13-s + (−0.309 − 0.951i)15-s + (−6.16 + 4.47i)17-s + (−2.38 + 7.33i)19-s + 0.763·21-s + 7.61·23-s + (0.309 − 0.951i)25-s + (−0.809 + 0.587i)27-s + (−0.809 − 2.48i)29-s + (1.11 + 0.812i)31-s + ⋯
L(s)  = 1  + (0.178 − 0.549i)3-s + (0.361 − 0.262i)5-s + (0.0892 + 0.274i)7-s + (−0.269 − 0.195i)9-s + (0.918 + 0.396i)11-s + (1.31 + 0.954i)13-s + (−0.0797 − 0.245i)15-s + (−1.49 + 1.08i)17-s + (−0.546 + 1.68i)19-s + 0.166·21-s + 1.58·23-s + (0.0618 − 0.190i)25-s + (−0.155 + 0.113i)27-s + (−0.150 − 0.462i)29-s + (0.200 + 0.145i)31-s + ⋯

Functional equation

Λ(s)=(1320s/2ΓC(s)L(s)=((0.9700.242i)Λ(2s)\begin{aligned}\Lambda(s)=\mathstrut & 1320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.970 - 0.242i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1320s/2ΓC(s+1/2)L(s)=((0.9700.242i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 1320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.970 - 0.242i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 13201320    =    2335112^{3} \cdot 3 \cdot 5 \cdot 11
Sign: 0.9700.242i0.970 - 0.242i
Analytic conductor: 10.540210.5402
Root analytic conductor: 3.246573.24657
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1320(961,)\chi_{1320} (961, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1320, ( :1/2), 0.9700.242i)(2,\ 1320,\ (\ :1/2),\ 0.970 - 0.242i)

Particular Values

L(1)L(1) \approx 1.9944463141.994446314
L(12)L(\frac12) \approx 1.9944463141.994446314
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=pFp(ps)1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1 1
3 1+(0.309+0.951i)T 1 + (-0.309 + 0.951i)T
5 1+(0.809+0.587i)T 1 + (-0.809 + 0.587i)T
11 1+(3.041.31i)T 1 + (-3.04 - 1.31i)T
good7 1+(0.2360.726i)T+(5.66+4.11i)T2 1 + (-0.236 - 0.726i)T + (-5.66 + 4.11i)T^{2}
13 1+(4.733.44i)T+(4.01+12.3i)T2 1 + (-4.73 - 3.44i)T + (4.01 + 12.3i)T^{2}
17 1+(6.164.47i)T+(5.2516.1i)T2 1 + (6.16 - 4.47i)T + (5.25 - 16.1i)T^{2}
19 1+(2.387.33i)T+(15.311.1i)T2 1 + (2.38 - 7.33i)T + (-15.3 - 11.1i)T^{2}
23 17.61T+23T2 1 - 7.61T + 23T^{2}
29 1+(0.809+2.48i)T+(23.4+17.0i)T2 1 + (0.809 + 2.48i)T + (-23.4 + 17.0i)T^{2}
31 1+(1.110.812i)T+(9.57+29.4i)T2 1 + (-1.11 - 0.812i)T + (9.57 + 29.4i)T^{2}
37 1+(0.972+2.99i)T+(29.9+21.7i)T2 1 + (0.972 + 2.99i)T + (-29.9 + 21.7i)T^{2}
41 1+(0.854+2.62i)T+(33.124.0i)T2 1 + (-0.854 + 2.62i)T + (-33.1 - 24.0i)T^{2}
43 1+3.32T+43T2 1 + 3.32T + 43T^{2}
47 1+(0.572+1.76i)T+(38.027.6i)T2 1 + (-0.572 + 1.76i)T + (-38.0 - 27.6i)T^{2}
53 1+(1.611.17i)T+(16.3+50.4i)T2 1 + (-1.61 - 1.17i)T + (16.3 + 50.4i)T^{2}
59 1+(0.3541.08i)T+(47.7+34.6i)T2 1 + (-0.354 - 1.08i)T + (-47.7 + 34.6i)T^{2}
61 1+(10.47.60i)T+(18.858.0i)T2 1 + (10.4 - 7.60i)T + (18.8 - 58.0i)T^{2}
67 115.5T+67T2 1 - 15.5T + 67T^{2}
71 1+(21.967.5i)T2 1 + (21.9 - 67.5i)T^{2}
73 1+(2.14+6.60i)T+(59.0+42.9i)T2 1 + (2.14 + 6.60i)T + (-59.0 + 42.9i)T^{2}
79 1+(5.784.20i)T+(24.4+75.1i)T2 1 + (-5.78 - 4.20i)T + (24.4 + 75.1i)T^{2}
83 1+(6.47+4.70i)T+(25.678.9i)T2 1 + (-6.47 + 4.70i)T + (25.6 - 78.9i)T^{2}
89 11.23T+89T2 1 - 1.23T + 89T^{2}
97 1+(9.09+6.60i)T+(29.9+92.2i)T2 1 + (9.09 + 6.60i)T + (29.9 + 92.2i)T^{2}
show more
show less
   L(s)=p j=12(1αj,pps)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

−9.390271085412432312460156755001, −8.763251849656354018566742316092, −8.332378963019535327018655792956, −7.00974946816908279233160614095, −6.41124158539066140422942580259, −5.75467499838648262005227939669, −4.36898806183121812968621525665, −3.69389939103109713504884390591, −2.08216086008798193308182354172, −1.42925092985306223727873298812, 0.907268419458868295573186266966, 2.58461704059945877190505257873, 3.41671770822027582616929991666, 4.50239760415966084056603939491, 5.26652005991497531524159193121, 6.49858090769500093112824254072, 6.88203164629428727461525623226, 8.194119159991469157814075266723, 9.108485065154399847544996347908, 9.236956770503022320061335512593

Graph of the ZZ-function along the critical line