# Properties

 Degree 4 Conductor $2^{2} \cdot 7^{2}$ Sign $1$ Motivic weight 3 Primitive no Self-dual yes Analytic rank 0

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 2·2-s + 3-s − 7·5-s + 2·6-s − 20·7-s − 8·8-s + 27·9-s − 14·10-s − 35·11-s + 132·13-s − 40·14-s − 7·15-s − 16·16-s − 59·17-s + 54·18-s − 137·19-s − 20·21-s − 70·22-s + 7·23-s − 8·24-s + 125·25-s + 264·26-s + 80·27-s + 212·29-s − 14·30-s − 75·31-s − 35·33-s + ⋯
 L(s)  = 1 + 0.707·2-s + 0.192·3-s − 0.626·5-s + 0.136·6-s − 1.07·7-s − 0.353·8-s + 9-s − 0.442·10-s − 0.959·11-s + 2.81·13-s − 0.763·14-s − 0.120·15-s − 1/4·16-s − 0.841·17-s + 0.707·18-s − 1.65·19-s − 0.207·21-s − 0.678·22-s + 0.0634·23-s − 0.0680·24-s + 25-s + 1.99·26-s + 0.570·27-s + 1.35·29-s − 0.0852·30-s − 0.434·31-s − 0.184·33-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$4$$ $$N$$ = $$196$$    =    $$2^{2} \cdot 7^{2}$$ $$\varepsilon$$ = $1$ motivic weight = $$3$$ character : induced by $\chi_{14} (1, \cdot )$ primitive : no self-dual : yes analytic rank = $$0$$ Selberg data = $$(4,\ 196,\ (\ :3/2, 3/2),\ 1)$$ $$L(2)$$ $$\approx$$ $$1.15357$$ $$L(\frac12)$$ $$\approx$$ $$1.15357$$ $$L(\frac{5}{2})$$ not available $$L(1)$$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{2,\;7\}$,$$F_p(T)$$ is a polynomial of degree 4. If $p \in \{2,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad2$C_2$ $$1 - p T + p^{2} T^{2}$$
7$C_2$ $$1 + 20 T + p^{3} T^{2}$$
good3$C_2^2$ $$1 - T - 26 T^{2} - p^{3} T^{3} + p^{6} T^{4}$$
5$C_2^2$ $$1 + 7 T - 76 T^{2} + 7 p^{3} T^{3} + p^{6} T^{4}$$
11$C_2^2$ $$1 + 35 T - 106 T^{2} + 35 p^{3} T^{3} + p^{6} T^{4}$$
13$C_2$ $$( 1 - 66 T + p^{3} T^{2} )^{2}$$
17$C_2^2$ $$1 + 59 T - 1432 T^{2} + 59 p^{3} T^{3} + p^{6} T^{4}$$
19$C_2^2$ $$1 + 137 T + 11910 T^{2} + 137 p^{3} T^{3} + p^{6} T^{4}$$
23$C_2^2$ $$1 - 7 T - 12118 T^{2} - 7 p^{3} T^{3} + p^{6} T^{4}$$
29$C_2$ $$( 1 - 106 T + p^{3} T^{2} )^{2}$$
31$C_2^2$ $$1 + 75 T - 24166 T^{2} + 75 p^{3} T^{3} + p^{6} T^{4}$$
37$C_2^2$ $$1 + 11 T - 50532 T^{2} + 11 p^{3} T^{3} + p^{6} T^{4}$$
41$C_2$ $$( 1 + 498 T + p^{3} T^{2} )^{2}$$
43$C_2$ $$( 1 - 260 T + p^{3} T^{2} )^{2}$$
47$C_2^2$ $$1 - 171 T - 74582 T^{2} - 171 p^{3} T^{3} + p^{6} T^{4}$$
53$C_2^2$ $$1 - 417 T + 25012 T^{2} - 417 p^{3} T^{3} + p^{6} T^{4}$$
59$C_2^2$ $$1 - 17 T - 205090 T^{2} - 17 p^{3} T^{3} + p^{6} T^{4}$$
61$C_2^2$ $$1 + 51 T - 224380 T^{2} + 51 p^{3} T^{3} + p^{6} T^{4}$$
67$C_2^2$ $$1 + 439 T - 108042 T^{2} + 439 p^{3} T^{3} + p^{6} T^{4}$$
71$C_2$ $$( 1 + 784 T + p^{3} T^{2} )^{2}$$
73$C_2^2$ $$1 + 295 T - 301992 T^{2} + 295 p^{3} T^{3} + p^{6} T^{4}$$
79$C_2^2$ $$1 - 495 T - 248014 T^{2} - 495 p^{3} T^{3} + p^{6} T^{4}$$
83$C_2$ $$( 1 - 932 T + p^{3} T^{2} )^{2}$$
89$C_2^2$ $$1 - 873 T + 57160 T^{2} - 873 p^{3} T^{3} + p^{6} T^{4}$$
97$C_2$ $$( 1 + 290 T + p^{3} T^{2} )^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}