Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{4} $
Sign $-1$
Motivic weight 5
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 9.76·5-s + 136.·7-s − 653.·11-s + 250.·13-s − 249.·17-s − 1.75e3·19-s + 1.65e3·23-s − 3.02e3·25-s − 4.24e3·29-s + 8.98e3·31-s + 1.33e3·35-s − 6.00e3·37-s + 1.07e4·41-s + 1.00e4·43-s − 2.34e4·47-s + 1.87e3·49-s − 9.41e3·53-s − 6.38e3·55-s − 4.41e4·59-s − 2.24e4·61-s + 2.44e3·65-s − 3.60e4·67-s − 7.85e4·71-s + 6.13e4·73-s − 8.92e4·77-s + 2.74e4·79-s + 6.48e4·83-s + ⋯
L(s)  = 1  + 0.174·5-s + 1.05·7-s − 1.62·11-s + 0.411·13-s − 0.209·17-s − 1.11·19-s + 0.652·23-s − 0.969·25-s − 0.937·29-s + 1.67·31-s + 0.184·35-s − 0.720·37-s + 0.998·41-s + 0.828·43-s − 1.55·47-s + 0.111·49-s − 0.460·53-s − 0.284·55-s − 1.65·59-s − 0.770·61-s + 0.0718·65-s − 0.979·67-s − 1.84·71-s + 1.34·73-s − 1.71·77-s + 0.495·79-s + 1.03·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(324\)    =    \(2^{2} \cdot 3^{4}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(5\)
character  :  $\chi_{324} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(1\)
Selberg data  =  \((2,\ 324,\ (\ :5/2),\ -1)\)
\(L(3)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{7}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{2,\;3\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 - 9.76T + 3.12e3T^{2} \)
7 \( 1 - 136.T + 1.68e4T^{2} \)
11 \( 1 + 653.T + 1.61e5T^{2} \)
13 \( 1 - 250.T + 3.71e5T^{2} \)
17 \( 1 + 249.T + 1.41e6T^{2} \)
19 \( 1 + 1.75e3T + 2.47e6T^{2} \)
23 \( 1 - 1.65e3T + 6.43e6T^{2} \)
29 \( 1 + 4.24e3T + 2.05e7T^{2} \)
31 \( 1 - 8.98e3T + 2.86e7T^{2} \)
37 \( 1 + 6.00e3T + 6.93e7T^{2} \)
41 \( 1 - 1.07e4T + 1.15e8T^{2} \)
43 \( 1 - 1.00e4T + 1.47e8T^{2} \)
47 \( 1 + 2.34e4T + 2.29e8T^{2} \)
53 \( 1 + 9.41e3T + 4.18e8T^{2} \)
59 \( 1 + 4.41e4T + 7.14e8T^{2} \)
61 \( 1 + 2.24e4T + 8.44e8T^{2} \)
67 \( 1 + 3.60e4T + 1.35e9T^{2} \)
71 \( 1 + 7.85e4T + 1.80e9T^{2} \)
73 \( 1 - 6.13e4T + 2.07e9T^{2} \)
79 \( 1 - 2.74e4T + 3.07e9T^{2} \)
83 \( 1 - 6.48e4T + 3.93e9T^{2} \)
89 \( 1 - 3.46e4T + 5.58e9T^{2} \)
97 \( 1 - 1.61e4T + 8.58e9T^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−10.60094467087870855400251550778, −9.356939044536823394773929127228, −8.207377920015571297224542618083, −7.72334893518644689185209611497, −6.30624198080997899257513307151, −5.24653281741732777704446569391, −4.36518321836775606719347977816, −2.76604282395064791780248206941, −1.64368616646693991397073991578, 0, 1.64368616646693991397073991578, 2.76604282395064791780248206941, 4.36518321836775606719347977816, 5.24653281741732777704446569391, 6.30624198080997899257513307151, 7.72334893518644689185209611497, 8.207377920015571297224542618083, 9.356939044536823394773929127228, 10.60094467087870855400251550778

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