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  + 26.3·5-s + 63.2·7-s − 98.2·11-s − 738.·13-s − 250.·17-s + 1.10e3·19-s − 4.40e3·23-s − 2.43e3·25-s + 7.88e3·29-s − 4.61e3·31-s + 1.66e3·35-s + 1.18e4·37-s − 1.00e4·41-s + 7.03e3·43-s − 1.49e4·47-s − 1.28e4·49-s − 2.24e4·53-s − 2.58e3·55-s − 1.08e4·59-s − 1.18e3·61-s − 1.94e4·65-s + 5.91e4·67-s − 1.43e4·71-s − 5.30e4·73-s − 6.21e3·77-s − 3.73e4·79-s − 1.20e5·83-s + ⋯
L(s)  = 1  + 0.470·5-s + 0.487·7-s − 0.244·11-s − 1.21·13-s − 0.209·17-s + 0.700·19-s − 1.73·23-s − 0.778·25-s + 1.74·29-s − 0.861·31-s + 0.229·35-s + 1.42·37-s − 0.936·41-s + 0.580·43-s − 0.985·47-s − 0.761·49-s − 1.09·53-s − 0.115·55-s − 0.404·59-s − 0.0409·61-s − 0.570·65-s + 1.61·67-s − 0.337·71-s − 1.16·73-s − 0.119·77-s − 0.674·79-s − 1.92·83-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 \( 1 - 26.3T + 3.12e3T^{2} \)
7 \( 1 - 63.2T + 1.68e4T^{2} \)
11 \( 1 + 98.2T + 1.61e5T^{2} \)
13 \( 1 + 738.T + 3.71e5T^{2} \)
17 \( 1 + 250.T + 1.41e6T^{2} \)
19 \( 1 - 1.10e3T + 2.47e6T^{2} \)
23 \( 1 + 4.40e3T + 6.43e6T^{2} \)
29 \( 1 - 7.88e3T + 2.05e7T^{2} \)
31 \( 1 + 4.61e3T + 2.86e7T^{2} \)
37 \( 1 - 1.18e4T + 6.93e7T^{2} \)
41 \( 1 + 1.00e4T + 1.15e8T^{2} \)
43 \( 1 - 7.03e3T + 1.47e8T^{2} \)
47 \( 1 + 1.49e4T + 2.29e8T^{2} \)
53 \( 1 + 2.24e4T + 4.18e8T^{2} \)
59 \( 1 + 1.08e4T + 7.14e8T^{2} \)
61 \( 1 + 1.18e3T + 8.44e8T^{2} \)
67 \( 1 - 5.91e4T + 1.35e9T^{2} \)
71 \( 1 + 1.43e4T + 1.80e9T^{2} \)
73 \( 1 + 5.30e4T + 2.07e9T^{2} \)
79 \( 1 + 3.73e4T + 3.07e9T^{2} \)
83 \( 1 + 1.20e5T + 3.93e9T^{2} \)
89 \( 1 + 9.78e4T + 5.58e9T^{2} \)
97 \( 1 - 1.06e5T + 8.58e9T^{2} \)
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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

−10.07173603264989808726576262842, −9.654834890446066644215057379993, −8.293899219943424710910718656811, −7.55123651571994127480931011585, −6.33901350729370476817138475984, −5.30056133150604752703753672681, −4.32213646864380957059259828185, −2.75846139199883843758015648619, −1.66023429767524299783915574209, 0, 1.66023429767524299783915574209, 2.75846139199883843758015648619, 4.32213646864380957059259828185, 5.30056133150604752703753672681, 6.33901350729370476817138475984, 7.55123651571994127480931011585, 8.293899219943424710910718656811, 9.654834890446066644215057379993, 10.07173603264989808726576262842

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