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Properties

Label 2-2268-63.32-c0-0-0
Degree $2$
Conductor $2268$
Sign $0.805 - 0.592i$
Analytic cond. $1.13187$
Root an. cond. $1.06389$
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)7-s + (0.5 − 0.866i)13-s + (0.5 + 0.866i)19-s + 25-s + (0.5 + 0.866i)31-s + (0.5 + 0.866i)37-s + (0.5 + 0.866i)43-s + (−0.499 − 0.866i)49-s + (−1 + 1.73i)61-s + (0.5 + 0.866i)67-s + (0.5 − 0.866i)73-s + (0.5 − 0.866i)79-s + (0.499 + 0.866i)91-s + (−1 − 1.73i)97-s − 103-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)7-s + (0.5 − 0.866i)13-s + (0.5 + 0.866i)19-s + 25-s + (0.5 + 0.866i)31-s + (0.5 + 0.866i)37-s + (0.5 + 0.866i)43-s + (−0.499 − 0.866i)49-s + (−1 + 1.73i)61-s + (0.5 + 0.866i)67-s + (0.5 − 0.866i)73-s + (0.5 − 0.866i)79-s + (0.499 + 0.866i)91-s + (−1 − 1.73i)97-s − 103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.805 - 0.592i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.805 - 0.592i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2268\)    =    \(2^{2} \cdot 3^{4} \cdot 7\)
Sign: $0.805 - 0.592i$
Analytic conductor: \(1.13187\)
Root analytic conductor: \(1.06389\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2268} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2268,\ (\ :0),\ 0.805 - 0.592i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.148744807\)
\(L(\frac12)\) \(\approx\) \(1.148744807\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 + (0.5 - 0.866i)T \)
good5 \( 1 - T^{2} \)
11 \( 1 - T^{2} \)
13 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
17 \( 1 + (0.5 + 0.866i)T^{2} \)
19 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 + (0.5 - 0.866i)T^{2} \)
31 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
37 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
41 \( 1 + (0.5 + 0.866i)T^{2} \)
43 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
47 \( 1 + (0.5 + 0.866i)T^{2} \)
53 \( 1 + (0.5 + 0.866i)T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
79 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 + (0.5 - 0.866i)T^{2} \)
89 \( 1 + (0.5 - 0.866i)T^{2} \)
97 \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \)
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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

−9.237170542814613976888280239677, −8.512769220661477125424529172716, −7.898944490705169190623942359637, −6.88843538891503310155864051112, −6.06835495951443995566578729167, −5.49192022219242850871640485841, −4.53395818778021085677674268190, −3.31613638099334255020726173121, −2.75592969845313402678990414357, −1.30718116133893840645938333017, 0.928506364829398284787484866633, 2.35650938725235909753171928929, 3.46109840750289883802906301299, 4.22654243073684989423628408290, 5.06999682094199822997274891697, 6.19318321187977936776921390043, 6.81285161015878071365300197384, 7.47935974545566306832226860597, 8.372166272236905830619959097365, 9.317780582646917971321759921381

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