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Properties

Label 2-3240-360.139-c0-0-13
Degree $2$
Conductor $3240$
Sign $0.173 + 0.984i$
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 + (−0.5 − 0.866i)7-s − 0.999·8-s + 0.999·10-s + (−1 − 1.73i)11-s + (−0.5 + 0.866i)13-s + (0.499 − 0.866i)14-s + (−0.5 − 0.866i)16-s − 19-s + (0.499 + 0.866i)20-s + (0.999 − 1.73i)22-s + (−0.5 + 0.866i)23-s + (−0.499 − 0.866i)25-s − 0.999·26-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)2-s + (−0.499 + 0.866i)4-s + (0.5 − 0.866i)5-s + (−0.5 − 0.866i)7-s − 0.999·8-s + 0.999·10-s + (−1 − 1.73i)11-s + (−0.5 + 0.866i)13-s + (0.499 − 0.866i)14-s + (−0.5 − 0.866i)16-s − 19-s + (0.499 + 0.866i)20-s + (0.999 − 1.73i)22-s + (−0.5 + 0.866i)23-s + (−0.499 − 0.866i)25-s − 0.999·26-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(3240\)    =    \(2^{3} \cdot 3^{4} \cdot 5\)
Sign: $0.173 + 0.984i$
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: \((2,\ 3240,\ (\ :0),\ 0.173 + 0.984i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7873538570\)
\(L(\frac12)\) \(\approx\) \(0.7873538570\)
\(L(1)\) 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 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \)
13 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
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 \( 1 + 2T + T^{2} \)
41 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
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 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
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 \( 1 - 2T + T^{2} \)
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

−8.592146186600905207199010545852, −7.932854904842347185063128532297, −7.05741951983430526880300858984, −6.36825306964191468156133545408, −5.57494243814410927230452997667, −5.06483125179744344767590084824, −4.02982758383732957212725432772, −3.43739111128036782504130111231, −2.15989324418934498254941759176, −0.35885198895837900450068662967, 2.01363870509096744759729799137, 2.45300861189565052257435052963, 3.17091040151054590340488845392, 4.36600969732535765400587024428, 5.13607577752474990027587761519, 5.83561951833600399530233948508, 6.58574643577698575444650269880, 7.38832894674501836911077391259, 8.401562012503533660608637083480, 9.300241446207699138752368806779

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