Properties

Label 2-2160-9.7-c1-0-21
Degree $2$
Conductor $2160$
Sign $-0.939 + 0.342i$
Analytic cond. $17.2476$
Root an. cond. $4.15303$
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 + (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

\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.939 + 0.342i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.939 + 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2160\)    =    \(2^{4} \cdot 3^{3} \cdot 5\)
Sign: $-0.939 + 0.342i$
Analytic conductor: \(17.2476\)
Root analytic conductor: \(4.15303\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2160} (721, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2160,\ (\ :1/2),\ -0.939 + 0.342i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6436018044\)
\(L(\frac12)\) \(\approx\) \(0.6436018044\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-2.5 + 4.33i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 3T + 17T^{2} \)
19 \( 1 + 5T + 19T^{2} \)
23 \( 1 + (3 + 5.19i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (5 - 8.66i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (1 + 1.73i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 4T + 37T^{2} \)
41 \( 1 + (1.5 + 2.59i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.5 + 2.59i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (2 - 3.46i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 6T + 53T^{2} \)
59 \( 1 + (-1.5 - 2.59i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1 - 1.73i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (5.5 + 9.52i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 14T + 71T^{2} \)
73 \( 1 + 15T + 73T^{2} \)
79 \( 1 + (-5 + 8.66i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-6 + 10.3i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 14T + 89T^{2} \)
97 \( 1 + (-6.5 + 11.2i)T + (-48.5 - 84.0i)T^{2} \)
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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

−8.940403965776297884536880497636, −8.114780567337784726864762637992, −7.14998900217653976309035900039, −6.28680717671166035417709363644, −5.70733891978350759841606202092, −4.51604435548213052849703621204, −3.91767374355500027577096924479, −2.84072250887378718078314778442, −1.59168597887098548294120467714, −0.21552206557671963973280089429, 1.69561167442075605649894026703, 2.53432958240295813330293143094, 3.98978339108529072480769504268, 4.26989429313363851772515879910, 5.52730855089390466044360912338, 6.42215607730218376178347788465, 7.06717807493524218520804241978, 7.78766142118267506199257065619, 8.655241520453248167470365463302, 9.527254866038541719902160716674

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