↑

Properties

Label 2-2160-9.7-c1-0-21
Degree 22
Conductor 21602160
Sign −0.939+0.342i-0.939 + 0.342i
Analytic cond. 17.247617.2476
Root an. cond. 4.153034.15303
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  + (−0.5 − 0.866i)5-s + (2.5 − 4.33i)11-s − 3·17-s − 5·19-s + (−3 − 5.19i)23-s + (−0.499 + 0.866i)25-s + (−5 + 8.66i)29-s + (−1 − 1.73i)31-s + 4·37-s + (−1.5 − 2.59i)41-s + (1.5 − 2.59i)43-s + (−2 + 3.46i)47-s + (3.5 + 6.06i)49-s + 6·53-s − 5·55-s + ⋯
L(s)  = 1  + (−0.223 − 0.387i)5-s + (0.753 − 1.30i)11-s − 0.727·17-s − 1.14·19-s + (−0.625 − 1.08i)23-s + (−0.0999 + 0.173i)25-s + (−0.928 + 1.60i)29-s + (−0.179 − 0.311i)31-s + 0.657·37-s + (−0.234 − 0.405i)41-s + (0.228 − 0.396i)43-s + (−0.291 + 0.505i)47-s + (0.5 + 0.866i)49-s + 0.824·53-s − 0.674·55-s + ⋯

Functional equation

Λ(s)=(2160s/2 ΓC(s) L(s)=((−0.939+0.342i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.939 + 0.342i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(2160s/2 ΓC(s+1/2) L(s)=((−0.939+0.342i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.939 + 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 21602160    =    24⋅33⋅52^{4} \cdot 3^{3} \cdot 5
Sign: −0.939+0.342i-0.939 + 0.342i
Analytic conductor: 17.247617.2476
Root analytic conductor: 4.153034.15303
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ2160(721,⋅)\chi_{2160} (721, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2160, ( :1/2), −0.939+0.342i)(2,\ 2160,\ (\ :1/2),\ -0.939 + 0.342i)

Particular Values

L(1)L(1) ≈\approx 0.64360180440.6436018044
L(12)L(\frac12) ≈\approx 0.64360180440.6436018044
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 1
3 1 1
5 1+(0.5+0.866i)T 1 + (0.5 + 0.866i)T
good7 1+(−3.5−6.06i)T2 1 + (-3.5 - 6.06i)T^{2}
11 1+(−2.5+4.33i)T+(−5.5−9.52i)T2 1 + (-2.5 + 4.33i)T + (-5.5 - 9.52i)T^{2}
13 1+(−6.5+11.2i)T2 1 + (-6.5 + 11.2i)T^{2}
17 1+3T+17T2 1 + 3T + 17T^{2}
19 1+5T+19T2 1 + 5T + 19T^{2}
23 1+(3+5.19i)T+(−11.5+19.9i)T2 1 + (3 + 5.19i)T + (-11.5 + 19.9i)T^{2}
29 1+(5−8.66i)T+(−14.5−25.1i)T2 1 + (5 - 8.66i)T + (-14.5 - 25.1i)T^{2}
31 1+(1+1.73i)T+(−15.5+26.8i)T2 1 + (1 + 1.73i)T + (-15.5 + 26.8i)T^{2}
37 1−4T+37T2 1 - 4T + 37T^{2}
41 1+(1.5+2.59i)T+(−20.5+35.5i)T2 1 + (1.5 + 2.59i)T + (-20.5 + 35.5i)T^{2}
43 1+(−1.5+2.59i)T+(−21.5−37.2i)T2 1 + (-1.5 + 2.59i)T + (-21.5 - 37.2i)T^{2}
47 1+(2−3.46i)T+(−23.5−40.7i)T2 1 + (2 - 3.46i)T + (-23.5 - 40.7i)T^{2}
53 1−6T+53T2 1 - 6T + 53T^{2}
59 1+(−1.5−2.59i)T+(−29.5+51.0i)T2 1 + (-1.5 - 2.59i)T + (-29.5 + 51.0i)T^{2}
61 1+(1−1.73i)T+(−30.5−52.8i)T2 1 + (1 - 1.73i)T + (-30.5 - 52.8i)T^{2}
67 1+(5.5+9.52i)T+(−33.5+58.0i)T2 1 + (5.5 + 9.52i)T + (-33.5 + 58.0i)T^{2}
71 1+14T+71T2 1 + 14T + 71T^{2}
73 1+15T+73T2 1 + 15T + 73T^{2}
79 1+(−5+8.66i)T+(−39.5−68.4i)T2 1 + (-5 + 8.66i)T + (-39.5 - 68.4i)T^{2}
83 1+(−6+10.3i)T+(−41.5−71.8i)T2 1 + (-6 + 10.3i)T + (-41.5 - 71.8i)T^{2}
89 1+14T+89T2 1 + 14T + 89T^{2}
97 1+(−6.5+11.2i)T+(−48.5−84.0i)T2 1 + (-6.5 + 11.2i)T + (-48.5 - 84.0i)T^{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

−8.940403965776297884536880497636, −8.114780567337784726864762637992, −7.14998900217653976309035900039, −6.28680717671166035417709363644, −5.70733891978350759841606202092, −4.51604435548213052849703621204, −3.91767374355500027577096924479, −2.84072250887378718078314778442, −1.59168597887098548294120467714, −0.21552206557671963973280089429, 1.69561167442075605649894026703, 2.53432958240295813330293143094, 3.98978339108529072480769504268, 4.26989429313363851772515879910, 5.52730855089390466044360912338, 6.42215607730218376178347788465, 7.06717807493524218520804241978, 7.78766142118267506199257065619, 8.655241520453248167470365463302, 9.527254866038541719902160716674

Graph of the ZZ-function along the critical line