# Properties

 Degree 2 Conductor $2^{2} \cdot 3^{3}$ Sign $-0.293 + 0.956i$ Motivic weight 3 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + (−4.40 + 2.76i)3-s + (−14.9 + 12.5i)5-s + (23.5 − 8.56i)7-s + (11.7 − 24.3i)9-s + (−39.5 − 33.1i)11-s + (−10.2 − 57.8i)13-s + (31.1 − 96.4i)15-s + (−3.52 − 6.09i)17-s + (−42.6 + 73.9i)19-s + (−79.9 + 102. i)21-s + (32.2 + 11.7i)23-s + (44.3 − 251. i)25-s + (15.3 + 139. i)27-s + (37.0 − 210. i)29-s + (−133. − 48.7i)31-s + ⋯
 L(s)  = 1 + (−0.847 + 0.531i)3-s + (−1.33 + 1.12i)5-s + (1.27 − 0.462i)7-s + (0.435 − 0.900i)9-s + (−1.08 − 0.909i)11-s + (−0.217 − 1.23i)13-s + (0.536 − 1.65i)15-s + (−0.0502 − 0.0869i)17-s + (−0.515 + 0.892i)19-s + (−0.830 + 1.06i)21-s + (0.292 + 0.106i)23-s + (0.354 − 2.01i)25-s + (0.109 + 0.994i)27-s + (0.237 − 1.34i)29-s + (−0.776 − 0.282i)31-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.293 + 0.956i)\, \overline{\Lambda}(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.293 + 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$108$$    =    $$2^{2} \cdot 3^{3}$$ $$\varepsilon$$ = $-0.293 + 0.956i$ motivic weight = $$3$$ character : $\chi_{108} (25, \cdot )$ primitive : yes self-dual : no analytic rank = $$0$$ Selberg data = $$(2,\ 108,\ (\ :3/2),\ -0.293 + 0.956i)$$ $$L(2)$$ $$\approx$$ $$0.201061 - 0.271989i$$ $$L(\frac12)$$ $$\approx$$ $$0.201061 - 0.271989i$$ $$L(\frac{5}{2})$$ 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 + (4.40 - 2.76i)T$$
good5 $$1 + (14.9 - 12.5i)T + (21.7 - 123. i)T^{2}$$
7 $$1 + (-23.5 + 8.56i)T + (262. - 220. i)T^{2}$$
11 $$1 + (39.5 + 33.1i)T + (231. + 1.31e3i)T^{2}$$
13 $$1 + (10.2 + 57.8i)T + (-2.06e3 + 751. i)T^{2}$$
17 $$1 + (3.52 + 6.09i)T + (-2.45e3 + 4.25e3i)T^{2}$$
19 $$1 + (42.6 - 73.9i)T + (-3.42e3 - 5.94e3i)T^{2}$$
23 $$1 + (-32.2 - 11.7i)T + (9.32e3 + 7.82e3i)T^{2}$$
29 $$1 + (-37.0 + 210. i)T + (-2.29e4 - 8.34e3i)T^{2}$$
31 $$1 + (133. + 48.7i)T + (2.28e4 + 1.91e4i)T^{2}$$
37 $$1 + (65.2 + 113. i)T + (-2.53e4 + 4.38e4i)T^{2}$$
41 $$1 + (8.52 + 48.3i)T + (-6.47e4 + 2.35e4i)T^{2}$$
43 $$1 + (-126. - 106. i)T + (1.38e4 + 7.82e4i)T^{2}$$
47 $$1 + (470. - 171. i)T + (7.95e4 - 6.67e4i)T^{2}$$
53 $$1 + 347.T + 1.48e5T^{2}$$
59 $$1 + (172. - 144. i)T + (3.56e4 - 2.02e5i)T^{2}$$
61 $$1 + (577. - 210. i)T + (1.73e5 - 1.45e5i)T^{2}$$
67 $$1 + (162. + 919. i)T + (-2.82e5 + 1.02e5i)T^{2}$$
71 $$1 + (-46.7 - 80.9i)T + (-1.78e5 + 3.09e5i)T^{2}$$
73 $$1 + (133. - 232. i)T + (-1.94e5 - 3.36e5i)T^{2}$$
79 $$1 + (155. - 883. i)T + (-4.63e5 - 1.68e5i)T^{2}$$
83 $$1 + (-6.17 + 35.0i)T + (-5.37e5 - 1.95e5i)T^{2}$$
89 $$1 + (-361. + 626. i)T + (-3.52e5 - 6.10e5i)T^{2}$$
97 $$1 + (1.21e3 + 1.01e3i)T + (1.58e5 + 8.98e5i)T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}