# Properties

 Degree 2 Conductor 2 Sign $-1$ Motivic weight 39 Primitive yes Self-dual yes Analytic rank 1

# Origins

## Dirichlet series

 L(s)  = 1 + 5.24e5·2-s − 7.35e8·3-s + 2.74e11·4-s − 1.62e13·5-s − 3.85e14·6-s + 1.60e16·7-s + 1.44e17·8-s − 3.51e18·9-s − 8.50e18·10-s − 1.67e20·11-s − 2.02e20·12-s − 1.32e21·13-s + 8.41e21·14-s + 1.19e22·15-s + 7.55e22·16-s − 4.96e23·17-s − 1.84e24·18-s − 1.14e25·19-s − 4.46e24·20-s − 1.18e25·21-s − 8.77e25·22-s − 6.66e26·23-s − 1.05e26·24-s − 1.55e27·25-s − 6.93e26·26-s + 5.56e27·27-s + 4.41e27·28-s + ⋯
 L(s)  = 1 + 0.707·2-s − 0.365·3-s + 1/2·4-s − 0.380·5-s − 0.258·6-s + 0.532·7-s + 0.353·8-s − 0.866·9-s − 0.269·10-s − 0.825·11-s − 0.182·12-s − 0.251·13-s + 0.376·14-s + 0.138·15-s + 1/4·16-s − 0.503·17-s − 0.612·18-s − 1.33·19-s − 0.190·20-s − 0.194·21-s − 0.583·22-s − 1.86·23-s − 0.129·24-s − 0.855·25-s − 0.177·26-s + 0.681·27-s + 0.266·28-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 2 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(40-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 2 ^{s/2} \, \Gamma_{\C}(s+39/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$2$$ $$\varepsilon$$ = $-1$ motivic weight = $$39$$ character : $\chi_{2} (1, \cdot )$ primitive : yes self-dual : yes analytic rank = $$1$$ Selberg data = $$(2,\ 2,\ (\ :39/2),\ -1)$$ $$L(20)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{41}{2})$$ not available $$L(1)$$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \neq 2$,$$F_p(T)$$ is a polynomial of degree 2. If $p = 2$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 $$1 - p^{19} T$$
good3 $$1 + 9079732 p^{4} T + p^{39} T^{2}$$
5 $$1 + 129809431866 p^{3} T + p^{39} T^{2}$$
7 $$1 - 2292866539412552 p T + p^{39} T^{2}$$
11 $$1 + 15220886148183602868 p T + p^{39} T^{2}$$
13 $$1 +$$$$10\!\cdots\!34$$$$p T + p^{39} T^{2}$$
17 $$1 +$$$$17\!\cdots\!94$$$$p^{2} T + p^{39} T^{2}$$
19 $$1 +$$$$60\!\cdots\!20$$$$p T + p^{39} T^{2}$$
23 $$1 +$$$$66\!\cdots\!72$$$$T + p^{39} T^{2}$$
29 $$1 -$$$$15\!\cdots\!50$$$$p T + p^{39} T^{2}$$
31 $$1 -$$$$15\!\cdots\!92$$$$T + p^{39} T^{2}$$
37 $$1 +$$$$69\!\cdots\!58$$$$p T + p^{39} T^{2}$$
41 $$1 -$$$$51\!\cdots\!42$$$$T + p^{39} T^{2}$$
43 $$1 -$$$$78\!\cdots\!28$$$$T + p^{39} T^{2}$$
47 $$1 -$$$$24\!\cdots\!04$$$$T + p^{39} T^{2}$$
53 $$1 -$$$$69\!\cdots\!98$$$$T + p^{39} T^{2}$$
59 $$1 +$$$$20\!\cdots\!00$$$$T + p^{39} T^{2}$$
61 $$1 +$$$$12\!\cdots\!18$$$$T + p^{39} T^{2}$$
67 $$1 -$$$$45\!\cdots\!64$$$$T + p^{39} T^{2}$$
71 $$1 +$$$$99\!\cdots\!28$$$$T + p^{39} T^{2}$$
73 $$1 -$$$$81\!\cdots\!18$$$$T + p^{39} T^{2}$$
79 $$1 +$$$$85\!\cdots\!40$$$$T + p^{39} T^{2}$$
83 $$1 -$$$$71\!\cdots\!48$$$$T + p^{39} T^{2}$$
89 $$1 +$$$$13\!\cdots\!90$$$$T + p^{39} T^{2}$$
97 $$1 -$$$$76\!\cdots\!34$$$$T + p^{39} T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}