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Properties

Label 2-1080-9.7-c1-0-10
Degree $2$
Conductor $1080$
Sign $-0.642 + 0.766i$
Analytic cond. $8.62384$
Root an. cond. $2.93663$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

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

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

Invariants

Degree: \(2\)
Conductor: \(1080\)    =    \(2^{3} \cdot 3^{3} \cdot 5\)
Sign: $-0.642 + 0.766i$
Analytic conductor: \(8.62384\)
Root analytic conductor: \(2.93663\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1080} (721, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1080,\ (\ :1/2),\ -0.642 + 0.766i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6239491700\)
\(L(\frac12)\) \(\approx\) \(0.6239491700\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 + (0.5 + 0.866i)T \)
good7 \( 1 + (1.86 - 3.23i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-2.73 + 4.73i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.732 + 1.26i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 7.46T + 17T^{2} \)
19 \( 1 + 2T + 19T^{2} \)
23 \( 1 + (-0.133 - 0.232i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-4.23 + 7.33i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-1 - 1.73i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 10.3T + 37T^{2} \)
41 \( 1 + (1.96 + 3.40i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-5.73 + 9.92i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-1.86 + 3.23i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 6T + 53T^{2} \)
59 \( 1 + (3.19 + 5.53i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.767 + 1.33i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.86 + 8.42i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 2.53T + 71T^{2} \)
73 \( 1 + 6.92T + 73T^{2} \)
79 \( 1 + (4.26 - 7.39i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (1.40 - 2.42i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 3.92T + 89T^{2} \)
97 \( 1 + (2.46 - 4.26i)T + (-48.5 - 84.0i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

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 $Z$-function along the critical line