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  + 81.4·5-s − 179.·7-s + 500.·11-s − 550.·13-s − 753.·17-s − 2.57e3·19-s + 2.74e3·23-s + 3.50e3·25-s − 3.90e3·29-s − 3.10e3·31-s − 1.46e4·35-s − 9.56e3·37-s − 2.22e3·41-s + 1.42e4·43-s + 6.47e3·47-s + 1.53e4·49-s − 1.36e4·53-s + 4.07e4·55-s − 5.70e3·59-s − 1.17e4·61-s − 4.48e4·65-s − 3.54e3·67-s + 5.84e4·71-s − 6.01e4·73-s − 8.97e4·77-s − 5.56e4·79-s − 3.99e4·83-s + ⋯
L(s)  = 1  + 1.45·5-s − 1.38·7-s + 1.24·11-s − 0.903·13-s − 0.632·17-s − 1.63·19-s + 1.08·23-s + 1.12·25-s − 0.863·29-s − 0.580·31-s − 2.01·35-s − 1.14·37-s − 0.206·41-s + 1.17·43-s + 0.427·47-s + 0.912·49-s − 0.669·53-s + 1.81·55-s − 0.213·59-s − 0.406·61-s − 1.31·65-s − 0.0964·67-s + 1.37·71-s − 1.32·73-s − 1.72·77-s − 1.00·79-s − 0.637·83-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 \( 1 - 81.4T + 3.12e3T^{2} \)
7 \( 1 + 179.T + 1.68e4T^{2} \)
11 \( 1 - 500.T + 1.61e5T^{2} \)
13 \( 1 + 550.T + 3.71e5T^{2} \)
17 \( 1 + 753.T + 1.41e6T^{2} \)
19 \( 1 + 2.57e3T + 2.47e6T^{2} \)
23 \( 1 - 2.74e3T + 6.43e6T^{2} \)
29 \( 1 + 3.90e3T + 2.05e7T^{2} \)
31 \( 1 + 3.10e3T + 2.86e7T^{2} \)
37 \( 1 + 9.56e3T + 6.93e7T^{2} \)
41 \( 1 + 2.22e3T + 1.15e8T^{2} \)
43 \( 1 - 1.42e4T + 1.47e8T^{2} \)
47 \( 1 - 6.47e3T + 2.29e8T^{2} \)
53 \( 1 + 1.36e4T + 4.18e8T^{2} \)
59 \( 1 + 5.70e3T + 7.14e8T^{2} \)
61 \( 1 + 1.17e4T + 8.44e8T^{2} \)
67 \( 1 + 3.54e3T + 1.35e9T^{2} \)
71 \( 1 - 5.84e4T + 1.80e9T^{2} \)
73 \( 1 + 6.01e4T + 2.07e9T^{2} \)
79 \( 1 + 5.56e4T + 3.07e9T^{2} \)
83 \( 1 + 3.99e4T + 3.93e9T^{2} \)
89 \( 1 + 1.03e5T + 5.58e9T^{2} \)
97 \( 1 + 1.65e5T + 8.58e9T^{2} \)
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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

−10.13694944966823491640330539469, −9.328541073977934098755543718295, −8.921743351185673944685577142258, −6.98817600846899891005321374232, −6.47197516409793372379963857212, −5.55569526743377612113979719676, −4.14217288434362424546825074333, −2.76873424366301355636633439915, −1.69923780458545297895026541371, 0, 1.69923780458545297895026541371, 2.76873424366301355636633439915, 4.14217288434362424546825074333, 5.55569526743377612113979719676, 6.47197516409793372379963857212, 6.98817600846899891005321374232, 8.921743351185673944685577142258, 9.328541073977934098755543718295, 10.13694944966823491640330539469

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