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Properties

Label 2-3240-360.149-c0-0-0
Degree $2$
Conductor $3240$
Sign $0.342 - 0.939i$
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  − i·2-s − 4-s + (−0.866 − 0.5i)5-s + i·8-s + (−0.5 + 0.866i)10-s + 16-s − 1.73·17-s − 1.73i·19-s + (0.866 + 0.5i)20-s + (−0.866 + 1.5i)23-s + (0.499 + 0.866i)25-s + (−0.5 + 0.866i)31-s − i·32-s + 1.73i·34-s − 1.73·38-s + ⋯
L(s)  = 1  − i·2-s − 4-s + (−0.866 − 0.5i)5-s + i·8-s + (−0.5 + 0.866i)10-s + 16-s − 1.73·17-s − 1.73i·19-s + (0.866 + 0.5i)20-s + (−0.866 + 1.5i)23-s + (0.499 + 0.866i)25-s + (−0.5 + 0.866i)31-s − i·32-s + 1.73i·34-s − 1.73·38-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.342 - 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.342 - 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3240\)    =    \(2^{3} \cdot 3^{4} \cdot 5\)
Sign: $0.342 - 0.939i$
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: \((2,\ 3240,\ (\ :0),\ 0.342 - 0.939i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1654240958\)
\(L(\frac12)\) \(\approx\) \(0.1654240958\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
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 \( 1 + 1.73T + T^{2} \)
19 \( 1 + 1.73iT - T^{2} \)
23 \( 1 + (0.866 - 1.5i)T + (-0.5 - 0.866i)T^{2} \)
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 \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (-0.5 - 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
83 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (0.5 - 0.866i)T^{2} \)
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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

−9.080712709409430810522135597110, −8.497352108277832721235756606329, −7.63267444094975218813804586757, −6.90310180334051216849122198532, −5.71870495705001337905000507628, −4.76729136112723748026856557436, −4.34291067170030829538784112380, −3.42827410943315165790413122365, −2.51874298260600094706205378746, −1.37662698929940882081737963097, 0.10261714209476857876601106088, 2.08953417827449260666885644185, 3.44670066560834599902806679822, 4.16091703119872284675289495023, 4.75253021460708638876513866141, 5.96353860121633907851240175810, 6.46042364327909233884324520573, 7.16801304898131059398958451575, 8.025920609314038408333503648660, 8.348464082515799315707530623489

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