# Properties

 Degree 4 Conductor $13 \cdot 29^{2}$ Sign $-1$ Motivic weight 1 Primitive yes Self-dual yes Analytic rank 1

# Origins

## Dirichlet series

 L(s)  = 1 − 4-s − 5-s − 6·7-s − 4·9-s + 5·13-s − 3·16-s + 20-s − 3·23-s − 3·25-s + 6·28-s − 29-s + 6·35-s + 4·36-s + 4·45-s + 13·49-s − 5·52-s − 8·53-s − 8·59-s + 24·63-s + 7·64-s − 5·65-s + 11·67-s + 20·71-s + 3·80-s + 7·81-s + 18·83-s − 30·91-s + ⋯
 L(s)  = 1 − 1/2·4-s − 0.447·5-s − 2.26·7-s − 4/3·9-s + 1.38·13-s − 3/4·16-s + 0.223·20-s − 0.625·23-s − 3/5·25-s + 1.13·28-s − 0.185·29-s + 1.01·35-s + 2/3·36-s + 0.596·45-s + 13/7·49-s − 0.693·52-s − 1.09·53-s − 1.04·59-s + 3.02·63-s + 7/8·64-s − 0.620·65-s + 1.34·67-s + 2.37·71-s + 0.335·80-s + 7/9·81-s + 1.97·83-s − 3.14·91-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 10933 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 10933 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$4$$ $$N$$ = $$10933$$    =    $$13 \cdot 29^{2}$$ $$\varepsilon$$ = $-1$ motivic weight = $$1$$ character : $\chi_{10933} (1, \cdot )$ Sato-Tate : $\mathrm{SU}(2)$ primitive : yes self-dual : yes analytic rank = 1 Selberg data = $(4,\ 10933,\ (\ :1/2, 1/2),\ -1)$ $L(1)$ $=$ $0$ $L(\frac12)$ $=$ $0$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \notin \{13,\;29\}$, $F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4$with $b_p = a_p^2 - a_{p^2}$. If $p \in \{13,\;29\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad13$C_1$$\times$$C_2$ $$( 1 - T )( 1 - 4 T + p T^{2} )$$
29$C_2$ $$1 + T + p T^{2}$$
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$$\times$$C_2$ $$( 1 - 2 T + p T^{2} )( 1 + 3 T + p T^{2} )$$
7$C_2$ $$( 1 + 3 T + p T^{2} )^{2}$$
11$C_2$ $$( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} )$$
17$C_2^2$ $$1 - 19 T^{2} + p^{2} T^{4}$$
19$C_2^2$ $$1 - 13 T^{2} + p^{2} T^{4}$$
23$C_2$$\times$$C_2$ $$( 1 - T + p T^{2} )( 1 + 4 T + p T^{2} )$$
31$C_2^2$ $$1 + 43 T^{2} + p^{2} T^{4}$$
37$C_2^2$ $$1 + 26 T^{2} + p^{2} T^{4}$$
41$C_2$ $$( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} )$$
43$C_2^2$ $$1 + 64 T^{2} + p^{2} T^{4}$$
47$C_2^2$ $$1 + 11 T^{2} + p^{2} T^{4}$$
53$C_2$$\times$$C_2$ $$( 1 - T + p T^{2} )( 1 + 9 T + p T^{2} )$$
59$C_2$$\times$$C_2$ $$( 1 - 6 T + p T^{2} )( 1 + 14 T + p T^{2} )$$
61$C_2^2$ $$1 - 37 T^{2} + p^{2} T^{4}$$
67$C_2$$\times$$C_2$ $$( 1 - 8 T + p T^{2} )( 1 - 3 T + p T^{2} )$$
71$C_2$$\times$$C_2$ $$( 1 - 15 T + p T^{2} )( 1 - 5 T + p T^{2} )$$
73$C_2^2$ $$1 + 104 T^{2} + p^{2} T^{4}$$
79$C_2^2$ $$1 + 32 T^{2} + p^{2} T^{4}$$
83$C_2$$\times$$C_2$ $$( 1 - 14 T + p T^{2} )( 1 - 4 T + p T^{2} )$$
89$C_2^2$ $$1 + 132 T^{2} + p^{2} T^{4}$$
97$C_2^2$ $$1 + 76 T^{2} + p^{2} T^{4}$$
\begin{aligned} L(s) = \prod_{\mathfrak{p}\ \mathrm{bad}} (1- a(\mathfrak{p}) (N\mathfrak{p})^{-s})^{-1} \prod_{\mathfrak{p}\ \mathrm{good}} (1- a(\mathfrak{p}) (N\mathfrak{p})^{-s} + (N\mathfrak{p})^{-2s})^{-1} \end{aligned}