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Properties

Label 2-3240-360.29-c0-0-3
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 − 0.999·8-s + 0.999·10-s + (−0.5 + 0.866i)11-s + (0.5 + 0.866i)13-s + (−0.5 + 0.866i)16-s + 17-s + (0.499 − 0.866i)20-s + (0.499 + 0.866i)22-s + (−0.5 − 0.866i)23-s + (−0.499 + 0.866i)25-s + 0.999·26-s + (−0.5 + 0.866i)29-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)2-s + (−0.499 − 0.866i)4-s + (0.5 + 0.866i)5-s − 0.999·8-s + 0.999·10-s + (−0.5 + 0.866i)11-s + (0.5 + 0.866i)13-s + (−0.5 + 0.866i)16-s + 17-s + (0.499 − 0.866i)20-s + (0.499 + 0.866i)22-s + (−0.5 − 0.866i)23-s + (−0.499 + 0.866i)25-s + 0.999·26-s + (−0.5 + 0.866i)29-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: $\chi_{3240} (269, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3240,\ (\ :0),\ 0.984 + 0.173i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.608174988\)
\(L(\frac12)\) \(\approx\) \(1.608174988\)
\(L(1)\) 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 \( 1 + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
13 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
17 \( 1 - T + T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
29 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
31 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
37 \( 1 - 2T + T^{2} \)
41 \( 1 + (0.5 - 0.866i)T^{2} \)
43 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
47 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
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

−9.164225326503075093526290437806, −8.111182062504364693057458488124, −7.16641169994682186097471998464, −6.40636365550386355958586814597, −5.74434651238776784845013659111, −4.85468627837880322833774618513, −4.04776543749108176063140305406, −3.12465080369670697783704529537, −2.34783654833587842096086494410, −1.46727638613897874483818129171, 0.892492290913744420472129927478, 2.55051594288242106292969631069, 3.50480834060103045059889593608, 4.34118311017043381265339896819, 5.30114540996283608005001692517, 5.83593196500109075263555327919, 6.21464575415869420849156469051, 7.61291398200573893463308492194, 8.023261178583268220212120078759, 8.526899519914217202915702190397

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