↑

Properties

Label 2-1080-9.4-c1-0-8
Degree 22
Conductor 10801080
Sign 0.173+0.984i0.173 + 0.984i
Analytic cond. 8.623848.62384
Root an. cond. 2.936632.93663
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)=(1080s/2 ΓC(s) L(s)=((0.173+0.984i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.173 + 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1080s/2 ΓC(s+1/2) L(s)=((0.173+0.984i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.173 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 10801080    =    23⋅33⋅52^{3} \cdot 3^{3} \cdot 5
Sign: 0.173+0.984i0.173 + 0.984i
Analytic conductor: 8.623848.62384
Root analytic conductor: 2.936632.93663
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1080(361,⋅)\chi_{1080} (361, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1080, ( :1/2), 0.173+0.984i)(2,\ 1080,\ (\ :1/2),\ 0.173 + 0.984i)

Particular Values

L(1)L(1) ≈\approx 1.1474813981.147481398
L(12)L(\frac12) ≈\approx 1.1474813981.147481398
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

−9.737554885136418086752983568606, −8.788706251432360872705215774460, −8.049922546873995106559747221301, −7.26834966634586408410464585410, −6.25184013947492097002411658442, −5.51186283950166454204954400442, −4.39603607525990205970093455395, −3.29675854143894482553301155332, −2.41506223502220968744004461809, −0.52786423627989410920559634578, 1.44122129871668389146920094671, 2.74230156776671143086075635522, 3.94209518255421705959660601965, 4.98099160613752284386215235684, 5.54365566509771538340622420165, 7.07016566737228500032490693960, 7.36291071449708716212659163074, 8.448724820006663224210405519239, 9.315041163704273039579873028536, 9.923563652643577875310518289806

Graph of the ZZ-function along the critical line