Properties

Label 2-2268-63.32-c0-0-1
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  + 7-s + (−1 + 1.73i)13-s + (0.5 + 0.866i)19-s + 25-s + (0.5 + 0.866i)31-s + (−1 − 1.73i)37-s + (0.5 + 0.866i)43-s + 49-s + (0.5 − 0.866i)61-s + (−1 − 1.73i)67-s + (0.5 − 0.866i)73-s + (−1 + 1.73i)79-s + (−1 + 1.73i)91-s + (0.5 + 0.866i)97-s + 2·103-s + ⋯
L(s)  = 1  + 7-s + (−1 + 1.73i)13-s + (0.5 + 0.866i)19-s + 25-s + (0.5 + 0.866i)31-s + (−1 − 1.73i)37-s + (0.5 + 0.866i)43-s + 49-s + (0.5 − 0.866i)61-s + (−1 − 1.73i)67-s + (0.5 − 0.866i)73-s + (−1 + 1.73i)79-s + (−1 + 1.73i)91-s + (0.5 + 0.866i)97-s + 2·103-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

\(L(\frac{1}{2})\) \(\approx\) \(1.277778509\)
\(L(\frac12)\) \(\approx\) \(1.277778509\)
\(L(1)\) 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 \( 1 - T \)
good5 \( 1 - T^{2} \)
11 \( 1 - T^{2} \)
13 \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \)
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 \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \)
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 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
79 \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 + (0.5 - 0.866i)T^{2} \)
89 \( 1 + (0.5 - 0.866i)T^{2} \)
97 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
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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

−9.188754919180011856069537541088, −8.611689903288101238030312326419, −7.64176372835243648768349244139, −7.10889829460470406674706521038, −6.22124977003612519667269099114, −5.11748946445603479251160522865, −4.62358964249994656542382993997, −3.65645765918656192194199195432, −2.35274301910678028761151887825, −1.49557311344948550455527650027, 0.982146521603076594537823915266, 2.42912079975785856482785934256, 3.18668971611496039336755154062, 4.53575872427349756876920266939, 5.10522681633471481915527360708, 5.81265083752014871343383155148, 7.04600569080584849253110280983, 7.57190484254547160774300182641, 8.359206202659616191092218338091, 8.965197137639307565490345864460

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