Properties

 Degree 4 Conductor $2^{2} \cdot 5^{2} \cdot 29$ Sign $1$ Motivic weight 1 Primitive yes Self-dual yes Analytic rank 0

Origins

Dirichlet series

 L(s)  = 1 − 4-s + 5-s − 11-s + 16-s − 5·19-s − 20-s − 4·25-s − 6·29-s − 31-s + 9·41-s + 44-s + 5·49-s − 55-s + 14·61-s − 64-s − 71-s + 5·76-s − 15·79-s + 80-s − 9·81-s + 15·89-s − 5·95-s + 4·100-s + 19·101-s + 20·109-s + 6·116-s − 15·121-s + ⋯
 L(s)  = 1 − 1/2·4-s + 0.447·5-s − 0.301·11-s + 1/4·16-s − 1.14·19-s − 0.223·20-s − 4/5·25-s − 1.11·29-s − 0.179·31-s + 1.40·41-s + 0.150·44-s + 5/7·49-s − 0.134·55-s + 1.79·61-s − 1/8·64-s − 0.118·71-s + 0.573·76-s − 1.68·79-s + 0.111·80-s − 81-s + 1.58·89-s − 0.512·95-s + 2/5·100-s + 1.89·101-s + 1.91·109-s + 0.557·116-s − 1.36·121-s + ⋯

Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 2900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 2900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

 $$d$$ = $$4$$ $$N$$ = $$2900$$    =    $$2^{2} \cdot 5^{2} \cdot 29$$ $$\varepsilon$$ = $1$ motivic weight = $$1$$ character : $\chi_{2900} (1, \cdot )$ Sato-Tate : $\mathrm{SU}(2)$ primitive : yes self-dual : yes analytic rank = 0 Selberg data = $(4,\ 2900,\ (\ :1/2, 1/2),\ 1)$ $L(1)$ $\approx$ $0.6834152878$ $L(\frac12)$ $\approx$ $0.6834152878$ $L(\frac{3}{2})$ not available $L(1)$ not available

Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \notin \{2,\;5,\;29\}$, $F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4$with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;5,\;29\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad2$C_2$ $$1 + T^{2}$$
5$C_2$ $$1 - T + p T^{2}$$
29$C_1$$\times$$C_2$ $$( 1 + T )( 1 + 5 T + p T^{2} )$$
good3$C_2^2$ $$1 + p^{2} T^{4}$$
7$C_2^2$ $$1 - 5 T^{2} + p^{2} T^{4}$$
11$C_2$$\times$$C_2$ $$( 1 - 2 T + p T^{2} )( 1 + 3 T + p T^{2} )$$
13$C_2^2$ $$1 + 15 T^{2} + p^{2} T^{4}$$
17$C_2^2$ $$1 - 20 T^{2} + p^{2} T^{4}$$
19$C_2$$\times$$C_2$ $$( 1 + p T^{2} )( 1 + 5 T + p T^{2} )$$
23$C_2$ $$( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )$$
31$C_2$$\times$$C_2$ $$( 1 - 7 T + p T^{2} )( 1 + 8 T + p T^{2} )$$
37$C_2^2$ $$1 - 40 T^{2} + p^{2} T^{4}$$
41$C_2$$\times$$C_2$ $$( 1 - 12 T + p T^{2} )( 1 + 3 T + p T^{2} )$$
43$C_2$ $$( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )$$
47$C_2^2$ $$1 + 50 T^{2} + p^{2} T^{4}$$
53$C_2$ $$( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )$$
59$C_2$ $$( 1 + p T^{2} )^{2}$$
61$C_2$$\times$$C_2$ $$( 1 - 12 T + p T^{2} )( 1 - 2 T + p T^{2} )$$
67$C_2^2$ $$1 - 5 T^{2} + p^{2} T^{4}$$
71$C_2$$\times$$C_2$ $$( 1 - 7 T + p T^{2} )( 1 + 8 T + p T^{2} )$$
73$C_2^2$ $$1 + 60 T^{2} + p^{2} T^{4}$$
79$C_2$$\times$$C_2$ $$( 1 + 5 T + p T^{2} )( 1 + 10 T + p T^{2} )$$
83$C_2^2$ $$1 + 15 T^{2} + p^{2} T^{4}$$
89$C_2$$\times$$C_2$ $$( 1 - 15 T + p T^{2} )( 1 + p T^{2} )$$
97$C_2^2$ $$1 + 150 T^{2} + p^{2} T^{4}$$
\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}