# Properties

 Degree $2$ Conductor $3$ Sign $1$ Motivic weight $66$ Primitive yes Self-dual yes Analytic rank $0$

# Origins

## Dirichlet series

 L(s)  = 1 − 5.55e15·3-s + 7.37e19·4-s − 1.54e28·7-s + 3.09e31·9-s − 4.10e35·12-s − 1.09e37·13-s + 5.44e39·16-s − 8.32e41·19-s + 8.59e43·21-s + 1.35e46·25-s − 1.71e47·27-s − 1.14e48·28-s − 2.08e49·31-s + 2.28e51·36-s − 7.41e51·37-s + 6.11e52·39-s − 1.24e54·43-s − 3.02e55·48-s + 1.79e56·49-s − 8.11e56·52-s + 4.62e57·57-s + 7.67e58·61-s − 4.77e59·63-s + 4.01e59·64-s + 2.36e59·67-s + 5.64e61·73-s − 7.53e61·75-s + ⋯
 L(s)  = 1 − 3-s + 4-s − 1.99·7-s + 9-s − 12-s − 1.90·13-s + 16-s − 0.526·19-s + 1.99·21-s + 25-s − 27-s − 1.99·28-s − 1.26·31-s + 36-s − 1.31·37-s + 1.90·39-s − 1.55·43-s − 48-s + 2.99·49-s − 1.90·52-s + 0.526·57-s + 0.931·61-s − 1.99·63-s + 64-s + 0.129·67-s + 1.82·73-s − 75-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(67-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+33) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$3$$ Sign: $1$ Motivic weight: $$66$$ Character: $\chi_{3} (2, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(2,\ 3,\ (\ :33),\ 1)$$

## Particular Values

 $$L(\frac{67}{2})$$ $$\approx$$ $$0.7517750513$$ $$L(\frac12)$$ $$\approx$$ $$0.7517750513$$ $$L(34)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1 + p^{33} T$$
good2 $$( 1 - p^{33} T )( 1 + p^{33} T )$$
5 $$( 1 - p^{33} T )( 1 + p^{33} T )$$
7 $$1 +$$$$15\!\cdots\!14$$$$T + p^{66} T^{2}$$
11 $$( 1 - p^{33} T )( 1 + p^{33} T )$$
13 $$1 +$$$$10\!\cdots\!06$$$$T + p^{66} T^{2}$$
17 $$( 1 - p^{33} T )( 1 + p^{33} T )$$
19 $$1 +$$$$83\!\cdots\!82$$$$T + p^{66} T^{2}$$
23 $$( 1 - p^{33} T )( 1 + p^{33} T )$$
29 $$( 1 - p^{33} T )( 1 + p^{33} T )$$
31 $$1 +$$$$20\!\cdots\!18$$$$T + p^{66} T^{2}$$
37 $$1 +$$$$74\!\cdots\!94$$$$T + p^{66} T^{2}$$
41 $$( 1 - p^{33} T )( 1 + p^{33} T )$$
43 $$1 +$$$$12\!\cdots\!86$$$$T + p^{66} T^{2}$$
47 $$( 1 - p^{33} T )( 1 + p^{33} T )$$
53 $$( 1 - p^{33} T )( 1 + p^{33} T )$$
59 $$( 1 - p^{33} T )( 1 + p^{33} T )$$
61 $$1 -$$$$76\!\cdots\!62$$$$T + p^{66} T^{2}$$
67 $$1 -$$$$23\!\cdots\!26$$$$T + p^{66} T^{2}$$
71 $$( 1 - p^{33} T )( 1 + p^{33} T )$$
73 $$1 -$$$$56\!\cdots\!34$$$$T + p^{66} T^{2}$$
79 $$1 -$$$$62\!\cdots\!78$$$$T + p^{66} T^{2}$$
83 $$( 1 - p^{33} T )( 1 + p^{33} T )$$
89 $$( 1 - p^{33} T )( 1 + p^{33} T )$$
97 $$1 -$$$$24\!\cdots\!46$$$$T + p^{66} T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$