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Properties

Label 2-1080-9.7-c1-0-10
Degree 22
Conductor 10801080
Sign −0.642+0.766i-0.642 + 0.766i
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 − 0.866i)5-s + (−1.86 + 3.23i)7-s + (2.73 − 4.73i)11-s + (−0.732 − 1.26i)13-s − 7.46·17-s − 2·19-s + (0.133 + 0.232i)23-s + (−0.499 + 0.866i)25-s + (4.23 − 7.33i)29-s + (1 + 1.73i)31-s + 3.73·35-s − 10.3·37-s + (−1.96 − 3.40i)41-s + (5.73 − 9.92i)43-s + (1.86 − 3.23i)47-s + ⋯
L(s)  = 1  + (−0.223 − 0.387i)5-s + (−0.705 + 1.22i)7-s + (0.823 − 1.42i)11-s + (−0.203 − 0.351i)13-s − 1.81·17-s − 0.458·19-s + (0.0279 + 0.0483i)23-s + (−0.0999 + 0.173i)25-s + (0.785 − 1.36i)29-s + (0.179 + 0.311i)31-s + 0.630·35-s − 1.70·37-s + (−0.306 − 0.531i)41-s + (0.874 − 1.51i)43-s + (0.272 − 0.471i)47-s + ⋯

Functional equation

Λ(s)=(1080s/2 ΓC(s) L(s)=((−0.642+0.766i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.642 + 0.766i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1080s/2 ΓC(s+1/2) L(s)=((−0.642+0.766i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.642 + 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 10801080    =    23⋅33⋅52^{3} \cdot 3^{3} \cdot 5
Sign: −0.642+0.766i-0.642 + 0.766i
Analytic conductor: 8.623848.62384
Root analytic conductor: 2.936632.93663
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1080(721,⋅)\chi_{1080} (721, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1080, ( :1/2), −0.642+0.766i)(2,\ 1080,\ (\ :1/2),\ -0.642 + 0.766i)

Particular Values

L(1)L(1) ≈\approx 0.62394917000.6239491700
L(12)L(\frac12) ≈\approx 0.62394917000.6239491700
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+(1.86−3.23i)T+(−3.5−6.06i)T2 1 + (1.86 - 3.23i)T + (-3.5 - 6.06i)T^{2}
11 1+(−2.73+4.73i)T+(−5.5−9.52i)T2 1 + (-2.73 + 4.73i)T + (-5.5 - 9.52i)T^{2}
13 1+(0.732+1.26i)T+(−6.5+11.2i)T2 1 + (0.732 + 1.26i)T + (-6.5 + 11.2i)T^{2}
17 1+7.46T+17T2 1 + 7.46T + 17T^{2}
19 1+2T+19T2 1 + 2T + 19T^{2}
23 1+(−0.133−0.232i)T+(−11.5+19.9i)T2 1 + (-0.133 - 0.232i)T + (-11.5 + 19.9i)T^{2}
29 1+(−4.23+7.33i)T+(−14.5−25.1i)T2 1 + (-4.23 + 7.33i)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+10.3T+37T2 1 + 10.3T + 37T^{2}
41 1+(1.96+3.40i)T+(−20.5+35.5i)T2 1 + (1.96 + 3.40i)T + (-20.5 + 35.5i)T^{2}
43 1+(−5.73+9.92i)T+(−21.5−37.2i)T2 1 + (-5.73 + 9.92i)T + (-21.5 - 37.2i)T^{2}
47 1+(−1.86+3.23i)T+(−23.5−40.7i)T2 1 + (-1.86 + 3.23i)T + (-23.5 - 40.7i)T^{2}
53 1+6T+53T2 1 + 6T + 53T^{2}
59 1+(3.19+5.53i)T+(−29.5+51.0i)T2 1 + (3.19 + 5.53i)T + (-29.5 + 51.0i)T^{2}
61 1+(−0.767+1.33i)T+(−30.5−52.8i)T2 1 + (-0.767 + 1.33i)T + (-30.5 - 52.8i)T^{2}
67 1+(4.86+8.42i)T+(−33.5+58.0i)T2 1 + (4.86 + 8.42i)T + (-33.5 + 58.0i)T^{2}
71 1+2.53T+71T2 1 + 2.53T + 71T^{2}
73 1+6.92T+73T2 1 + 6.92T + 73T^{2}
79 1+(4.26−7.39i)T+(−39.5−68.4i)T2 1 + (4.26 - 7.39i)T + (-39.5 - 68.4i)T^{2}
83 1+(1.40−2.42i)T+(−41.5−71.8i)T2 1 + (1.40 - 2.42i)T + (-41.5 - 71.8i)T^{2}
89 1+3.92T+89T2 1 + 3.92T + 89T^{2}
97 1+(2.46−4.26i)T+(−48.5−84.0i)T2 1 + (2.46 - 4.26i)T + (-48.5 - 84.0i)T^{2}
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   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

−9.262335065073245434679554905090, −8.800841032470035177337475043023, −8.285992034444571829552907365256, −6.83705245089892691540218865825, −6.19260814852499841284979275302, −5.41357006252683573814325483531, −4.24335343207621674679514631466, −3.21750272487700664190464266937, −2.14573448771341936091070718137, −0.26708532578271374096514364939, 1.63568201241005898827871350411, 2.99607191168199366347394650917, 4.26584033104075387218673488810, 4.53209715680901776536111674003, 6.32687809797563987872703316573, 6.93366075905501565320244440845, 7.31039571917350120949383216461, 8.659583141626784836386454775798, 9.418917396316539074774174283799, 10.23142241934004645927209797399

Graph of the ZZ-function along the critical line