# Properties

 Degree 2 Conductor 3 Sign $1$ Motivic weight 42 Primitive yes Self-dual yes Analytic rank 0

# Origins

## Dirichlet series

 L(s)  = 1 − 1.04e10·3-s + 4.39e12·4-s + 1.46e17·7-s + 1.09e20·9-s − 4.60e22·12-s − 3.57e23·13-s + 1.93e25·16-s − 1.65e26·19-s − 1.52e27·21-s + 2.27e29·25-s − 1.14e30·27-s + 6.43e29·28-s + 4.00e31·31-s + 4.81e32·36-s + 1.62e33·37-s + 3.73e33·39-s + 3.01e34·43-s − 2.02e35·48-s − 2.90e35·49-s − 1.57e36·52-s + 1.72e36·57-s + 5.58e37·61-s + 1.60e37·63-s + 8.50e37·64-s − 3.97e38·67-s − 1.16e39·73-s − 2.37e39·75-s + ⋯
 L(s)  = 1 − 3-s + 4-s + 0.261·7-s + 9-s − 12-s − 1.44·13-s + 16-s − 0.231·19-s − 0.261·21-s + 25-s − 27-s + 0.261·28-s + 1.92·31-s + 36-s + 1.90·37-s + 1.44·39-s + 1.50·43-s − 48-s − 0.931·49-s − 1.44·52-s + 0.231·57-s + 1.80·61-s + 0.261·63-s + 64-s − 1.78·67-s − 0.866·73-s − 75-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(43-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+21) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$3$$ $$\varepsilon$$ = $1$ motivic weight = $$42$$ character : $\chi_{3} (2, \cdot )$ primitive : yes self-dual : yes analytic rank = 0 Selberg data = $(2,\ 3,\ (\ :21),\ 1)$ $L(\frac{43}{2})$ $\approx$ $1.846569382$ $L(\frac12)$ $\approx$ $1.846569382$ $L(22)$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \neq 3$, $$F_p$$ is a polynomial of degree 2. If $p = 3$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad3 $$1 + p^{21} T$$
good2 $$( 1 - p^{21} T )( 1 + p^{21} T )$$
5 $$( 1 - p^{21} T )( 1 + p^{21} T )$$
7 $$1 - 146246101081752386 T + p^{42} T^{2}$$
11 $$( 1 - p^{21} T )( 1 + p^{21} T )$$
13 $$1 +$$$$35\!\cdots\!26$$$$T + p^{42} T^{2}$$
17 $$( 1 - p^{21} T )( 1 + p^{21} T )$$
19 $$1 +$$$$16\!\cdots\!62$$$$T + p^{42} T^{2}$$
23 $$( 1 - p^{21} T )( 1 + p^{21} T )$$
29 $$( 1 - p^{21} T )( 1 + p^{21} T )$$
31 $$1 -$$$$40\!\cdots\!62$$$$T + p^{42} T^{2}$$
37 $$1 -$$$$16\!\cdots\!26$$$$T + p^{42} T^{2}$$
41 $$( 1 - p^{21} T )( 1 + p^{21} T )$$
43 $$1 -$$$$30\!\cdots\!14$$$$T + p^{42} T^{2}$$
47 $$( 1 - p^{21} T )( 1 + p^{21} T )$$
53 $$( 1 - p^{21} T )( 1 + p^{21} T )$$
59 $$( 1 - p^{21} T )( 1 + p^{21} T )$$
61 $$1 -$$$$55\!\cdots\!22$$$$T + p^{42} T^{2}$$
67 $$1 +$$$$39\!\cdots\!34$$$$T + p^{42} T^{2}$$
71 $$( 1 - p^{21} T )( 1 + p^{21} T )$$
73 $$1 +$$$$11\!\cdots\!46$$$$T + p^{42} T^{2}$$
79 $$1 -$$$$14\!\cdots\!58$$$$T + p^{42} T^{2}$$
83 $$( 1 - p^{21} T )( 1 + p^{21} T )$$
89 $$( 1 - p^{21} T )( 1 + p^{21} T )$$
97 $$1 -$$$$22\!\cdots\!06$$$$T + p^{42} T^{2}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}