# Properties

 Degree 4 Conductor $7 \cdot 29^{2}$ Sign $1$ Motivic weight 1 Primitive yes Self-dual yes Analytic rank 0

# Origins

## Dirichlet series

 L(s)  = 1 − 4-s + 4·5-s + 7-s − 4·9-s − 2·13-s − 3·16-s − 4·20-s + 2·23-s + 2·25-s − 28-s + 4·29-s + 4·35-s + 4·36-s − 16·45-s − 4·49-s + 2·52-s + 12·53-s + 2·59-s − 4·63-s + 7·64-s − 8·65-s − 14·67-s − 12·80-s + 7·81-s − 12·83-s − 2·91-s − 2·92-s + ⋯
 L(s)  = 1 − 1/2·4-s + 1.78·5-s + 0.377·7-s − 4/3·9-s − 0.554·13-s − 3/4·16-s − 0.894·20-s + 0.417·23-s + 2/5·25-s − 0.188·28-s + 0.742·29-s + 0.676·35-s + 2/3·36-s − 2.38·45-s − 4/7·49-s + 0.277·52-s + 1.64·53-s + 0.260·59-s − 0.503·63-s + 7/8·64-s − 0.992·65-s − 1.71·67-s − 1.34·80-s + 7/9·81-s − 1.31·83-s − 0.209·91-s − 0.208·92-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 5887 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 5887 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$4$$ $$N$$ = $$5887$$    =    $$7 \cdot 29^{2}$$ $$\varepsilon$$ = $1$ motivic weight = $$1$$ character : $\chi_{5887} (1, \cdot )$ Sato-Tate : $\mathrm{SU}(2)$ primitive : yes self-dual : yes analytic rank = 0 Selberg data = $(4,\ 5887,\ (\ :1/2, 1/2),\ 1)$ $L(1)$ $\approx$ $0.9419312153$ $L(\frac12)$ $\approx$ $0.9419312153$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{7,\;29\}$,$F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4$with $b_p = a_p^2 - a_{p^2}$. If $p \in \{7,\;29\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad7$C_1$$\times$$C_2$ $$( 1 + T )( 1 - 2 T + p T^{2} )$$
29$C_2$ $$1 - 4 T + p T^{2}$$
good2$C_2^2$ $$1 + T^{2} + p^{2} T^{4}$$
3$C_2^2$ $$1 + 4 T^{2} + p^{2} T^{4}$$
5$C_2$ $$( 1 - 2 T + p T^{2} )^{2}$$
11$C_2$ $$( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )$$
13$C_2$ $$( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} )$$
17$C_2^2$ $$1 + 6 T^{2} + p^{2} T^{4}$$
19$C_2^2$ $$1 + 12 T^{2} + p^{2} T^{4}$$
23$C_2$$\times$$C_2$ $$( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} )$$
31$C_2^2$ $$1 - 32 T^{2} + p^{2} T^{4}$$
37$C_2^2$ $$1 + 26 T^{2} + p^{2} T^{4}$$
41$C_2^2$ $$1 - 2 T^{2} + p^{2} T^{4}$$
43$C_2^2$ $$1 - 26 T^{2} + p^{2} T^{4}$$
47$C_2^2$ $$1 - 44 T^{2} + p^{2} T^{4}$$
53$C_2$ $$( 1 - 6 T + p T^{2} )^{2}$$
59$C_2$$\times$$C_2$ $$( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} )$$
61$C_2$ $$( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} )$$
67$C_2$$\times$$C_2$ $$( 1 + 2 T + p T^{2} )( 1 + 12 T + p T^{2} )$$
71$C_2$ $$( 1 + p T^{2} )^{2}$$
73$C_2^2$ $$1 + 54 T^{2} + p^{2} T^{4}$$
79$C_2^2$ $$1 - 58 T^{2} + p^{2} T^{4}$$
83$C_2$$\times$$C_2$ $$( 1 - 4 T + p T^{2} )( 1 + 16 T + p T^{2} )$$
89$C_2$ $$( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )$$
97$C_2^2$ $$1 - 154 T^{2} + p^{2} T^{4}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}