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  − 110.·5-s + 101.·7-s + 150.·11-s + 635.·13-s − 1.49e3·17-s + 1.43e3·19-s + 1.26e3·23-s + 9.06e3·25-s + 2.77e3·29-s − 6.96e3·31-s − 1.12e4·35-s − 7.95e3·37-s + 2.02e3·41-s − 1.25e4·43-s + 6.48e3·47-s − 6.45e3·49-s − 9.82e3·53-s − 1.65e4·55-s − 4.70e4·59-s + 8.33e3·61-s − 7.01e4·65-s − 7.26e3·67-s − 3.58e3·71-s + 5.80e4·73-s + 1.52e4·77-s − 6.37e4·79-s − 8.28e4·83-s + ⋯
L(s)  = 1  − 1.97·5-s + 0.784·7-s + 0.374·11-s + 1.04·13-s − 1.25·17-s + 0.913·19-s + 0.498·23-s + 2.90·25-s + 0.613·29-s − 1.30·31-s − 1.54·35-s − 0.954·37-s + 0.188·41-s − 1.03·43-s + 0.428·47-s − 0.384·49-s − 0.480·53-s − 0.739·55-s − 1.76·59-s + 0.286·61-s − 2.05·65-s − 0.197·67-s − 0.0843·71-s + 1.27·73-s + 0.293·77-s − 1.14·79-s − 1.32·83-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 \( 1 + 110.T + 3.12e3T^{2} \)
7 \( 1 - 101.T + 1.68e4T^{2} \)
11 \( 1 - 150.T + 1.61e5T^{2} \)
13 \( 1 - 635.T + 3.71e5T^{2} \)
17 \( 1 + 1.49e3T + 1.41e6T^{2} \)
19 \( 1 - 1.43e3T + 2.47e6T^{2} \)
23 \( 1 - 1.26e3T + 6.43e6T^{2} \)
29 \( 1 - 2.77e3T + 2.05e7T^{2} \)
31 \( 1 + 6.96e3T + 2.86e7T^{2} \)
37 \( 1 + 7.95e3T + 6.93e7T^{2} \)
41 \( 1 - 2.02e3T + 1.15e8T^{2} \)
43 \( 1 + 1.25e4T + 1.47e8T^{2} \)
47 \( 1 - 6.48e3T + 2.29e8T^{2} \)
53 \( 1 + 9.82e3T + 4.18e8T^{2} \)
59 \( 1 + 4.70e4T + 7.14e8T^{2} \)
61 \( 1 - 8.33e3T + 8.44e8T^{2} \)
67 \( 1 + 7.26e3T + 1.35e9T^{2} \)
71 \( 1 + 3.58e3T + 1.80e9T^{2} \)
73 \( 1 - 5.80e4T + 2.07e9T^{2} \)
79 \( 1 + 6.37e4T + 3.07e9T^{2} \)
83 \( 1 + 8.28e4T + 3.93e9T^{2} \)
89 \( 1 - 3.86e3T + 5.58e9T^{2} \)
97 \( 1 - 6.92e4T + 8.58e9T^{2} \)
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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

−10.78105193034527534185270428006, −9.020273794704055464165657332739, −8.387221332764334098030246939231, −7.53076926224106409458648056011, −6.67294852820168709232195406607, −5.02496035666943270180765860902, −4.12674446149982694823352759070, −3.24357623964346825309420156213, −1.32859678886697448740980562983, 0, 1.32859678886697448740980562983, 3.24357623964346825309420156213, 4.12674446149982694823352759070, 5.02496035666943270180765860902, 6.67294852820168709232195406607, 7.53076926224106409458648056011, 8.387221332764334098030246939231, 9.020273794704055464165657332739, 10.78105193034527534185270428006

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