Properties

 Degree 2 Conductor $2^{2} \cdot 3^{3}$ Sign $1$ Motivic weight 4 Primitive yes Self-dual yes Analytic rank 0

Origins

Dirichlet series

 L(s)  = 1 + 23·7-s + 191·13-s + 647·19-s + 625·25-s + 194·31-s + 2.59e3·37-s − 3.21e3·43-s − 1.87e3·49-s − 5.23e3·61-s − 8.80e3·67-s + 9.79e3·73-s − 1.23e4·79-s + 4.39e3·91-s + 9.74e3·97-s + 3.43e3·103-s + 2.20e4·109-s + ⋯
 L(s)  = 1 + 0.469·7-s + 1.13·13-s + 1.79·19-s + 25-s + 0.201·31-s + 1.89·37-s − 1.73·43-s − 0.779·49-s − 1.40·61-s − 1.96·67-s + 1.83·73-s − 1.98·79-s + 0.530·91-s + 1.03·97-s + 0.323·103-s + 1.85·109-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

 $$d$$ = $$2$$ $$N$$ = $$108$$    =    $$2^{2} \cdot 3^{3}$$ $$\varepsilon$$ = $1$ motivic weight = $$4$$ character : $\chi_{108} (53, \cdot )$ primitive : yes self-dual : yes analytic rank = $$0$$ Selberg data = $$(2,\ 108,\ (\ :2),\ 1)$$ $$L(\frac{5}{2})$$ $$\approx$$ $$1.94008$$ $$L(\frac12)$$ $$\approx$$ $$1.94008$$ $$L(3)$$ 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 - p^{2} T )( 1 + p^{2} T )$$
7 $$1 - 23 T + p^{4} T^{2}$$
11 $$( 1 - p^{2} T )( 1 + p^{2} T )$$
13 $$1 - 191 T + p^{4} T^{2}$$
17 $$( 1 - p^{2} T )( 1 + p^{2} T )$$
19 $$1 - 647 T + p^{4} T^{2}$$
23 $$( 1 - p^{2} T )( 1 + p^{2} T )$$
29 $$( 1 - p^{2} T )( 1 + p^{2} T )$$
31 $$1 - 194 T + p^{4} T^{2}$$
37 $$1 - 2591 T + p^{4} T^{2}$$
41 $$( 1 - p^{2} T )( 1 + p^{2} T )$$
43 $$1 + 3214 T + p^{4} T^{2}$$
47 $$( 1 - p^{2} T )( 1 + p^{2} T )$$
53 $$( 1 - p^{2} T )( 1 + p^{2} T )$$
59 $$( 1 - p^{2} T )( 1 + p^{2} T )$$
61 $$1 + 5233 T + p^{4} T^{2}$$
67 $$1 + 8809 T + p^{4} T^{2}$$
71 $$( 1 - p^{2} T )( 1 + p^{2} T )$$
73 $$1 - 9791 T + p^{4} T^{2}$$
79 $$1 + 12361 T + p^{4} T^{2}$$
83 $$( 1 - p^{2} T )( 1 + p^{2} T )$$
89 $$( 1 - p^{2} T )( 1 + p^{2} T )$$
97 $$1 - 9743 T + p^{4} T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}