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Properties

Label 2-3240-360.149-c0-0-6
Degree $2$
Conductor $3240$
Sign $0.642 + 0.766i$
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 + (−0.866 − 0.5i)5-s + 0.999i·8-s + 0.999·10-s + (−0.5 − 0.866i)16-s + (−0.866 + 0.499i)20-s + (0.499 + 0.866i)25-s + (1 − 1.73i)31-s + (0.866 + 0.499i)32-s + (0.499 − 0.866i)40-s + (−0.5 + 0.866i)49-s + (−0.866 − 0.499i)50-s − 2i·53-s + 1.99i·62-s + ⋯
L(s)  = 1  + (−0.866 + 0.5i)2-s + (0.499 − 0.866i)4-s + (−0.866 − 0.5i)5-s + 0.999i·8-s + 0.999·10-s + (−0.5 − 0.866i)16-s + (−0.866 + 0.499i)20-s + (0.499 + 0.866i)25-s + (1 − 1.73i)31-s + (0.866 + 0.499i)32-s + (0.499 − 0.866i)40-s + (−0.5 + 0.866i)49-s + (−0.866 − 0.499i)50-s − 2i·53-s + 1.99i·62-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.642 + 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.642 + 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3240\)    =    \(2^{3} \cdot 3^{4} \cdot 5\)
Sign: $0.642 + 0.766i$
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: \((2,\ 3240,\ (\ :0),\ 0.642 + 0.766i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5910927790\)
\(L(\frac12)\) \(\approx\) \(0.5910927790\)
\(L(1)\) 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 \( 1 + (0.866 + 0.5i)T \)
good7 \( 1 + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (-0.5 + 0.866i)T^{2} \)
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 \( 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2} \)
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 \( 1 + 2iT - T^{2} \)
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 \( 1 - T^{2} \)
79 \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \)
83 \( 1 + (-1.73 + i)T + (0.5 - 0.866i)T^{2} \)
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

−8.600925695891779863173240780107, −7.996272892636752116304802902696, −7.48667078841001455594891293130, −6.62440239884931162313703003311, −5.87003467611588911861929575266, −4.96048539432965552089821410646, −4.22157391322116805781436794947, −3.05265151293774369865331767331, −1.84098176108223502377780476247, −0.55760815819448973741562244514, 1.12773261308307439439960525632, 2.46802424945646971584766763128, 3.23404979164300647388189657271, 4.00821449824052655356223141718, 4.93875542232301005363437512584, 6.27787453700765483036636592113, 6.91575639119992941301427770039, 7.58208190333492808125701338051, 8.287870561042720494781042928684, 8.825739051676924761881631201705

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