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Properties

Label 2-3240-360.139-c0-0-8
Degree $2$
Conductor $3240$
Sign $0.984 - 0.173i$
Analytic cond. $1.61697$
Root an. cond. $1.27160$
Motivic weight $0$
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)2-s + (−0.499 + 0.866i)4-s + (−0.5 + 0.866i)5-s + (−0.866 − 1.5i)7-s − 0.999·8-s − 0.999·10-s + (0.866 − 1.5i)13-s + (0.866 − 1.5i)14-s + (−0.5 − 0.866i)16-s + 19-s + (−0.499 − 0.866i)20-s + (0.5 − 0.866i)23-s + (−0.499 − 0.866i)25-s + 1.73·26-s + 1.73·28-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)2-s + (−0.499 + 0.866i)4-s + (−0.5 + 0.866i)5-s + (−0.866 − 1.5i)7-s − 0.999·8-s − 0.999·10-s + (0.866 − 1.5i)13-s + (0.866 − 1.5i)14-s + (−0.5 − 0.866i)16-s + 19-s + (−0.499 − 0.866i)20-s + (0.5 − 0.866i)23-s + (−0.499 − 0.866i)25-s + 1.73·26-s + 1.73·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 - 0.173i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 - 0.173i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3240\)    =    \(2^{3} \cdot 3^{4} \cdot 5\)
Sign: $0.984 - 0.173i$
Analytic conductor: \(1.61697\)
Root analytic conductor: \(1.27160\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3240} (2539, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3240,\ (\ :0),\ 0.984 - 0.173i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.133218275\)
\(L(\frac12)\) \(\approx\) \(1.133218275\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 - 0.866i)T \)
3 \( 1 \)
5 \( 1 + (0.5 - 0.866i)T \)
good7 \( 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (-0.5 + 0.866i)T^{2} \)
13 \( 1 + (-0.866 + 1.5i)T + (-0.5 - 0.866i)T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 - T + T^{2} \)
23 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
29 \( 1 + (0.5 - 0.866i)T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 + (0.866 - 1.5i)T + (-0.5 - 0.866i)T^{2} \)
43 \( 1 + (0.5 - 0.866i)T^{2} \)
47 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
53 \( 1 - T + T^{2} \)
59 \( 1 + (-0.866 + 1.5i)T + (-0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 + (0.5 - 0.866i)T^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + (0.5 - 0.866i)T^{2} \)
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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

−8.440141881838975367669910361928, −7.943808631577083191219521221964, −7.17698620491703867095965847443, −6.73426592358770549721278594892, −6.04372377298089296831350980265, −5.07618742596241310451295553329, −4.07888448322293493499599687612, −3.37360130882184617615265770753, −3.00325823765524994853220143996, −0.64456102762588714623166089734, 1.29749878035475756638985621608, 2.28833564812594222910012118577, 3.35572637876718864453464566369, 3.94792461803233146100748264192, 4.99667121621096878622761177034, 5.57882517531912244987344220843, 6.27805449952383456352392961837, 7.23185305379653742853246318242, 8.541533710053545533892675181234, 9.016243273875462987067090667165

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