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Properties

Label 2-1080-40.29-c1-0-22
Degree $2$
Conductor $1080$
Sign $0.978 + 0.207i$
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  + (0.393 − 1.35i)2-s + (−1.68 − 1.06i)4-s + (−1.33 − 1.79i)5-s + 4.27i·7-s + (−2.11 + 1.87i)8-s + (−2.96 + 1.10i)10-s + 0.381i·11-s + 3.13·13-s + (5.80 + 1.68i)14-s + (1.71 + 3.61i)16-s − 3.34i·17-s + 3.69i·19-s + (0.326 + 4.46i)20-s + (0.518 + 0.150i)22-s + 8.20i·23-s + ⋯
L(s)  = 1  + (0.278 − 0.960i)2-s + (−0.844 − 0.534i)4-s + (−0.595 − 0.803i)5-s + 1.61i·7-s + (−0.749 + 0.662i)8-s + (−0.937 + 0.347i)10-s + 0.115i·11-s + 0.868·13-s + (1.55 + 0.450i)14-s + (0.427 + 0.903i)16-s − 0.812i·17-s + 0.846i·19-s + (0.0730 + 0.997i)20-s + (0.110 + 0.0320i)22-s + 1.71i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.978 + 0.207i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.978 + 0.207i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1080\)    =    \(2^{3} \cdot 3^{3} \cdot 5\)
Sign: $0.978 + 0.207i$
Analytic conductor: \(8.62384\)
Root analytic conductor: \(2.93663\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1080} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1080,\ (\ :1/2),\ 0.978 + 0.207i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.323144479\)
\(L(\frac12)\) \(\approx\) \(1.323144479\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.393 + 1.35i)T \)
3 \( 1 \)
5 \( 1 + (1.33 + 1.79i)T \)
good7 \( 1 - 4.27iT - 7T^{2} \)
11 \( 1 - 0.381iT - 11T^{2} \)
13 \( 1 - 3.13T + 13T^{2} \)
17 \( 1 + 3.34iT - 17T^{2} \)
19 \( 1 - 3.69iT - 19T^{2} \)
23 \( 1 - 8.20iT - 23T^{2} \)
29 \( 1 + 2.98iT - 29T^{2} \)
31 \( 1 + 2.30T + 31T^{2} \)
37 \( 1 - 10.5T + 37T^{2} \)
41 \( 1 - 4.98T + 41T^{2} \)
43 \( 1 - 5.59T + 43T^{2} \)
47 \( 1 - 5.88iT - 47T^{2} \)
53 \( 1 - 4.40T + 53T^{2} \)
59 \( 1 + 5.51iT - 59T^{2} \)
61 \( 1 - 6.32iT - 61T^{2} \)
67 \( 1 + 12.6T + 67T^{2} \)
71 \( 1 - 15.7T + 71T^{2} \)
73 \( 1 + 1.92iT - 73T^{2} \)
79 \( 1 + 12.3T + 79T^{2} \)
83 \( 1 + 1.18T + 83T^{2} \)
89 \( 1 + 11.8T + 89T^{2} \)
97 \( 1 - 7.71iT - 97T^{2} \)
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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

−9.490148621745460366642514861120, −9.354375131514761434566290593558, −8.424480013198586620674552401752, −7.69917281165506460455268379914, −5.91018405042494455000258284197, −5.55956342664951553271680380465, −4.49127977216218517526524553665, −3.55366990502359995576781801130, −2.46583653645929372272848932673, −1.24233743507634755884120030953, 0.63344164019357881693455238361, 2.97979216072028262316991663796, 4.06097175265017528554853778442, 4.37706773982252127110221315818, 5.92248413127041450138478298590, 6.70691989378705558191479128572, 7.24856858887617184066349673161, 8.046372025878762096711478958738, 8.758595610793664789378488320361, 9.965625030439454045161288645219

Graph of the $Z$-function along the critical line