# Properties

 Degree 4 Conductor $2^{2} \cdot 5^{2}$ Sign $1$ Motivic weight 3 Primitive no Self-dual yes Analytic rank 0

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 4·4-s − 10·5-s + 50·9-s − 56·11-s + 16·16-s + 120·19-s + 40·20-s − 25·25-s − 180·29-s − 256·31-s − 200·36-s + 484·41-s + 224·44-s − 500·45-s + 10·49-s + 560·55-s + 40·59-s + 1.08e3·61-s − 64·64-s − 2.25e3·71-s − 480·76-s + 1.44e3·79-s − 160·80-s + 1.77e3·81-s + 980·89-s − 1.20e3·95-s − 2.80e3·99-s + ⋯
 L(s)  = 1 − 1/2·4-s − 0.894·5-s + 1.85·9-s − 1.53·11-s + 1/4·16-s + 1.44·19-s + 0.447·20-s − 1/5·25-s − 1.15·29-s − 1.48·31-s − 0.925·36-s + 1.84·41-s + 0.767·44-s − 1.65·45-s + 0.0291·49-s + 1.37·55-s + 0.0882·59-s + 2.27·61-s − 1/8·64-s − 3.77·71-s − 0.724·76-s + 2.05·79-s − 0.223·80-s + 2.42·81-s + 1.16·89-s − 1.29·95-s − 2.84·99-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$4$$ $$N$$ = $$100$$    =    $$2^{2} \cdot 5^{2}$$ $$\varepsilon$$ = $1$ motivic weight = $$3$$ character : induced by $\chi_{10} (1, \cdot )$ primitive : no self-dual : yes analytic rank = 0 Selberg data = $(4,\ 100,\ (\ :3/2, 3/2),\ 1)$ $L(2)$ $\approx$ $0.664284$ $L(\frac12)$ $\approx$ $0.664284$ $L(\frac{5}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \notin \{2,\;5\}$, $$F_p(T)$$ is a polynomial of degree 4. If $p \in \{2,\;5\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad2$C_2$ $$1 + p^{2} T^{2}$$
5$C_2$ $$1 + 2 p T + p^{3} T^{2}$$
good3$C_2^2$ $$1 - 50 T^{2} + p^{6} T^{4}$$
7$C_2^2$ $$1 - 10 T^{2} + p^{6} T^{4}$$
11$C_2$ $$( 1 + 28 T + p^{3} T^{2} )^{2}$$
13$C_2^2$ $$1 - 4250 T^{2} + p^{6} T^{4}$$
17$C_2^2$ $$1 - 5730 T^{2} + p^{6} T^{4}$$
19$C_2$ $$( 1 - 60 T + p^{3} T^{2} )^{2}$$
23$C_2^2$ $$1 - 20970 T^{2} + p^{6} T^{4}$$
29$C_2$ $$( 1 + 90 T + p^{3} T^{2} )^{2}$$
31$C_2$ $$( 1 + 128 T + p^{3} T^{2} )^{2}$$
37$C_2^2$ $$1 - 45610 T^{2} + p^{6} T^{4}$$
41$C_2$ $$( 1 - 242 T + p^{3} T^{2} )^{2}$$
43$C_2^2$ $$1 - 27970 T^{2} + p^{6} T^{4}$$
47$C_2^2$ $$1 - 156570 T^{2} + p^{6} T^{4}$$
53$C_2^2$ $$1 - 286090 T^{2} + p^{6} T^{4}$$
59$C_2$ $$( 1 - 20 T + p^{3} T^{2} )^{2}$$
61$C_2$ $$( 1 - 542 T + p^{3} T^{2} )^{2}$$
67$C_2^2$ $$1 - 413170 T^{2} + p^{6} T^{4}$$
71$C_2$ $$( 1 + 1128 T + p^{3} T^{2} )^{2}$$
73$C_2^2$ $$1 - 378610 T^{2} + p^{6} T^{4}$$
79$C_2$ $$( 1 - 720 T + p^{3} T^{2} )^{2}$$
83$C_2^2$ $$1 - 915090 T^{2} + p^{6} T^{4}$$
89$C_2$ $$( 1 - 490 T + p^{3} T^{2} )^{2}$$
97$C_2^2$ $$1 + 294590 T^{2} + p^{6} T^{4}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}