Properties

Label 2-18e2-4.3-c0-0-1
Degree $2$
Conductor $324$
Sign $1$
Analytic cond. $0.161697$
Root an. cond. $0.402115$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s − 5-s + 8-s − 10-s − 13-s + 16-s − 17-s − 20-s − 26-s − 29-s + 32-s − 34-s − 37-s − 40-s + 2·41-s + 49-s − 52-s + 2·53-s − 58-s − 61-s + 64-s + 65-s − 68-s − 73-s − 74-s − 80-s + ⋯
L(s)  = 1  + 2-s + 4-s − 5-s + 8-s − 10-s − 13-s + 16-s − 17-s − 20-s − 26-s − 29-s + 32-s − 34-s − 37-s − 40-s + 2·41-s + 49-s − 52-s + 2·53-s − 58-s − 61-s + 64-s + 65-s − 68-s − 73-s − 74-s − 80-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(324\)    =    \(2^{2} \cdot 3^{4}\)
Sign: $1$
Analytic conductor: \(0.161697\)
Root analytic conductor: \(0.402115\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{324} (163, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 324,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.188200343\)
\(L(\frac12)\) \(\approx\) \(1.188200343\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−11.92087871252312978719466341079, −11.22600475906675974031770849257, −10.31038993583302778273367538430, −8.954948356116674124248041963203, −7.65216242953380728411403903154, −7.10118567986892139403294393430, −5.80122382318090507880480519818, −4.62590085768480636130122278343, −3.78882632007711462473940508430, −2.39124768407718997212617200088, 2.39124768407718997212617200088, 3.78882632007711462473940508430, 4.62590085768480636130122278343, 5.80122382318090507880480519818, 7.10118567986892139403294393430, 7.65216242953380728411403903154, 8.954948356116674124248041963203, 10.31038993583302778273367538430, 11.22600475906675974031770849257, 11.92087871252312978719466341079

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