Properties

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

Origins

Dirichlet series

 L(s)  = 1 − 37·7-s − 19·13-s − 163·19-s − 125·25-s + 308·31-s + 323·37-s − 520·43-s + 1.02e3·49-s + 719·61-s − 127·67-s − 919·73-s − 1.38e3·79-s + 703·91-s − 523·97-s − 1.80e3·103-s − 646·109-s + ⋯
 L(s)  = 1 − 1.99·7-s − 0.405·13-s − 1.96·19-s − 25-s + 1.78·31-s + 1.43·37-s − 1.84·43-s + 2.99·49-s + 1.50·61-s − 0.231·67-s − 1.47·73-s − 1.97·79-s + 0.809·91-s − 0.547·97-s − 1.72·103-s − 0.567·109-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

 $$d$$ = $$2$$ $$N$$ = $$108$$    =    $$2^{2} \cdot 3^{3}$$ $$\varepsilon$$ = $-1$ motivic weight = $$3$$ character : $\chi_{108} (1, \cdot )$ primitive : yes self-dual : yes analytic rank = $$1$$ Selberg data = $$(2,\ 108,\ (\ :3/2),\ -1)$$ $$L(2)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{5}{2})$$ 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 + p^{3} T^{2}$$
7 $$1 + 37 T + p^{3} T^{2}$$
11 $$1 + p^{3} T^{2}$$
13 $$1 + 19 T + p^{3} T^{2}$$
17 $$1 + p^{3} T^{2}$$
19 $$1 + 163 T + p^{3} T^{2}$$
23 $$1 + p^{3} T^{2}$$
29 $$1 + p^{3} T^{2}$$
31 $$1 - 308 T + p^{3} T^{2}$$
37 $$1 - 323 T + p^{3} T^{2}$$
41 $$1 + p^{3} T^{2}$$
43 $$1 + 520 T + p^{3} T^{2}$$
47 $$1 + p^{3} T^{2}$$
53 $$1 + p^{3} T^{2}$$
59 $$1 + p^{3} T^{2}$$
61 $$1 - 719 T + p^{3} T^{2}$$
67 $$1 + 127 T + p^{3} T^{2}$$
71 $$1 + p^{3} T^{2}$$
73 $$1 + 919 T + p^{3} T^{2}$$
79 $$1 + 1387 T + p^{3} T^{2}$$
83 $$1 + p^{3} T^{2}$$
89 $$1 + p^{3} T^{2}$$
97 $$1 + 523 T + p^{3} T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}