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Properties

Label 2-720-9.7-c1-0-6
Degree 22
Conductor 720720
Sign 0.939−0.342i0.939 - 0.342i
Analytic cond. 5.749225.74922
Root an. cond. 2.397752.39775
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  + (−0.866 − 1.5i)3-s + (0.5 + 0.866i)5-s + (0.133 − 0.232i)7-s + (−1.5 + 2.59i)9-s + (−0.732 + 1.26i)11-s + (2.73 + 4.73i)13-s + (0.866 − 1.5i)15-s + 0.535·17-s + 2·19-s − 0.464·21-s + (1.86 + 3.23i)23-s + (−0.499 + 0.866i)25-s + 5.19·27-s + (−0.767 + 1.33i)29-s + (−1 − 1.73i)31-s + ⋯
L(s)  = 1  + (−0.499 − 0.866i)3-s + (0.223 + 0.387i)5-s + (0.0506 − 0.0877i)7-s + (−0.5 + 0.866i)9-s + (−0.220 + 0.382i)11-s + (0.757 + 1.31i)13-s + (0.223 − 0.387i)15-s + 0.129·17-s + 0.458·19-s − 0.101·21-s + (0.389 + 0.673i)23-s + (−0.0999 + 0.173i)25-s + 1.00·27-s + (−0.142 + 0.246i)29-s + (−0.179 − 0.311i)31-s + ⋯

Functional equation

Λ(s)=(720s/2 ΓC(s) L(s)=((0.939−0.342i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 - 0.342i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(720s/2 ΓC(s+1/2) L(s)=((0.939−0.342i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{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: 720720    =    24⋅32⋅52^{4} \cdot 3^{2} \cdot 5
Sign: 0.939−0.342i0.939 - 0.342i
Analytic conductor: 5.749225.74922
Root analytic conductor: 2.397752.39775
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ720(241,⋅)\chi_{720} (241, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 720, ( :1/2), 0.939−0.342i)(2,\ 720,\ (\ :1/2),\ 0.939 - 0.342i)

Particular Values

L(1)L(1) ≈\approx 1.25456+0.221213i1.25456 + 0.221213i
L(12)L(\frac12) ≈\approx 1.25456+0.221213i1.25456 + 0.221213i
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+(0.866+1.5i)T 1 + (0.866 + 1.5i)T
5 1+(−0.5−0.866i)T 1 + (-0.5 - 0.866i)T
good7 1+(−0.133+0.232i)T+(−3.5−6.06i)T2 1 + (-0.133 + 0.232i)T + (-3.5 - 6.06i)T^{2}
11 1+(0.732−1.26i)T+(−5.5−9.52i)T2 1 + (0.732 - 1.26i)T + (-5.5 - 9.52i)T^{2}
13 1+(−2.73−4.73i)T+(−6.5+11.2i)T2 1 + (-2.73 - 4.73i)T + (-6.5 + 11.2i)T^{2}
17 1−0.535T+17T2 1 - 0.535T + 17T^{2}
19 1−2T+19T2 1 - 2T + 19T^{2}
23 1+(−1.86−3.23i)T+(−11.5+19.9i)T2 1 + (-1.86 - 3.23i)T + (-11.5 + 19.9i)T^{2}
29 1+(0.767−1.33i)T+(−14.5−25.1i)T2 1 + (0.767 - 1.33i)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+(4.96+8.59i)T+(−20.5+35.5i)T2 1 + (4.96 + 8.59i)T + (-20.5 + 35.5i)T^{2}
43 1+(2.26−3.92i)T+(−21.5−37.2i)T2 1 + (2.26 - 3.92i)T + (-21.5 - 37.2i)T^{2}
47 1+(−0.133+0.232i)T+(−23.5−40.7i)T2 1 + (-0.133 + 0.232i)T + (-23.5 - 40.7i)T^{2}
53 1−6T+53T2 1 - 6T + 53T^{2}
59 1+(−7.19−12.4i)T+(−29.5+51.0i)T2 1 + (-7.19 - 12.4i)T + (-29.5 + 51.0i)T^{2}
61 1+(−4.23+7.33i)T+(−30.5−52.8i)T2 1 + (-4.23 + 7.33i)T + (-30.5 - 52.8i)T^{2}
67 1+(−3.13−5.42i)T+(−33.5+58.0i)T2 1 + (-3.13 - 5.42i)T + (-33.5 + 58.0i)T^{2}
71 1+9.46T+71T2 1 + 9.46T + 71T^{2}
73 1−6.92T+73T2 1 - 6.92T + 73T^{2}
79 1+(−7.73+13.3i)T+(−39.5−68.4i)T2 1 + (-7.73 + 13.3i)T + (-39.5 - 68.4i)T^{2}
83 1+(6.59−11.4i)T+(−41.5−71.8i)T2 1 + (6.59 - 11.4i)T + (-41.5 - 71.8i)T^{2}
89 1+9.92T+89T2 1 + 9.92T + 89T^{2}
97 1+(−4.46+7.73i)T+(−48.5−84.0i)T2 1 + (-4.46 + 7.73i)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

−10.64678491763587008843098802905, −9.613256421081321748046623447947, −8.700922513128986030203641919452, −7.59991870314145153478055969415, −6.97179043446640638886547515686, −6.13113912867951698656711326412, −5.25155443306023009544516508255, −3.98767178447564564844249093783, −2.50941293676367291269323027473, −1.35318532116602086919910884241, 0.805421824636803346274993369744, 2.88065150988372629488412583131, 3.89745579472869260738067587030, 5.09109234301985860271388624928, 5.66276724249526166519175194963, 6.59447129674172733634572766983, 8.038074580991079178119216828698, 8.671602372428560625602676072133, 9.675256627148733638606678687370, 10.33561388236564536529115086111

Graph of the ZZ-function along the critical line