Properties

Label 2-3240-360.149-c0-0-1
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  + (−0.866 + 0.5i)2-s + (0.499 − 0.866i)4-s + i·5-s + (−1.5 + 0.866i)7-s + 0.999i·8-s + (−0.5 − 0.866i)10-s + (0.866 + 1.5i)11-s + (0.866 − 1.5i)14-s + (−0.5 − 0.866i)16-s + (0.866 + 0.499i)20-s + (−1.5 − 0.866i)22-s − 25-s + 1.73i·28-s + (−0.5 + 0.866i)31-s + (0.866 + 0.499i)32-s + ⋯
L(s)  = 1  + (−0.866 + 0.5i)2-s + (0.499 − 0.866i)4-s + i·5-s + (−1.5 + 0.866i)7-s + 0.999i·8-s + (−0.5 − 0.866i)10-s + (0.866 + 1.5i)11-s + (0.866 − 1.5i)14-s + (−0.5 − 0.866i)16-s + (0.866 + 0.499i)20-s + (−1.5 − 0.866i)22-s − 25-s + 1.73i·28-s + (−0.5 + 0.866i)31-s + (0.866 + 0.499i)32-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: $\chi_{3240} (1349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3240,\ (\ :0),\ -0.984 + 0.173i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4843291544\)
\(L(\frac12)\) \(\approx\) \(0.4843291544\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 - 0.5i)T \)
3 \( 1 \)
5 \( 1 - iT \)
good7 \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (-0.866 - 1.5i)T + (-0.5 + 0.866i)T^{2} \)
13 \( 1 + (-0.5 - 0.866i)T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + (-0.5 - 0.866i)T^{2} \)
29 \( 1 + (-0.5 + 0.866i)T^{2} \)
31 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 + (0.5 + 0.866i)T^{2} \)
43 \( 1 + (-0.5 + 0.866i)T^{2} \)
47 \( 1 + (-0.5 + 0.866i)T^{2} \)
53 \( 1 - iT - T^{2} \)
59 \( 1 + (-0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (-0.5 - 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + 1.73iT - T^{2} \)
79 \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \)
83 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \)
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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

−9.235887207062972935586183831175, −8.704143181023695114344950674924, −7.41464089695222748062317455496, −7.11429201532736504961588342065, −6.31455066889641975786640723585, −5.95581874623056940509346025420, −4.77245836565835057012469384930, −3.53057802818439534571186404402, −2.64834891123815559122198046429, −1.78172900377901998307765621580, 0.40647121658180763381791473717, 1.30856039284869759166106914948, 2.77831347492752530463874381871, 3.72505414499907743636741295182, 4.06001028304231606187879510294, 5.58867255151892384532535292959, 6.37306760552634735358640744377, 6.98871521771987500328748156157, 7.940114340340254437234857569771, 8.606627446992557482131334599180

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