# Properties

 Degree 4 Conductor $11^{2}$ Sign $1$ Motivic weight 4 Primitive no Self-dual yes Analytic rank 0

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 6·3-s + 2·4-s + 62·5-s − 135·9-s + 22·11-s − 12·12-s − 372·15-s − 252·16-s + 124·20-s + 554·23-s + 1.63e3·25-s + 1.35e3·27-s − 2.72e3·31-s − 132·33-s − 270·36-s + 334·37-s + 44·44-s − 8.37e3·45-s + 3.40e3·47-s + 1.51e3·48-s + 1.80e3·49-s + 9.04e3·53-s + 1.36e3·55-s − 4.72e3·59-s − 744·60-s − 1.01e3·64-s − 5.60e3·67-s + ⋯
 L(s)  = 1 − 2/3·3-s + 1/8·4-s + 2.47·5-s − 5/3·9-s + 2/11·11-s − 0.0833·12-s − 1.65·15-s − 0.984·16-s + 0.309·20-s + 1.04·23-s + 2.61·25-s + 1.85·27-s − 2.83·31-s − 0.121·33-s − 0.208·36-s + 0.243·37-s + 1/44·44-s − 4.13·45-s + 1.54·47-s + 0.656·48-s + 0.750·49-s + 3.21·53-s + 0.450·55-s − 1.35·59-s − 0.206·60-s − 0.248·64-s − 1.24·67-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\C}(s+2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$4$$ $$N$$ = $$121$$    =    $$11^{2}$$ $$\varepsilon$$ = $1$ motivic weight = $$4$$ character : induced by $\chi_{11} (1, \cdot )$ primitive : no self-dual : yes analytic rank = 0 Selberg data = $(4,\ 121,\ (\ :2, 2),\ 1)$ $L(\frac{5}{2})$ $\approx$ $1.15767$ $L(\frac12)$ $\approx$ $1.15767$ $L(3)$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \neq 11$, $$F_p$$ is a polynomial of degree 4. If $p = 11$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad11$C_2$ $$1 - 2 p T + p^{4} T^{2}$$
good2$C_2^2$ $$1 - p T^{2} + p^{8} T^{4}$$
3$C_2$ $$( 1 + p T + p^{4} T^{2} )^{2}$$
5$C_2$ $$( 1 - 31 T + p^{4} T^{2} )^{2}$$
7$C_2^2$ $$1 - 1802 T^{2} + p^{8} T^{4}$$
13$C_2^2$ $$1 - 22442 T^{2} + p^{8} T^{4}$$
17$C_2^2$ $$1 - 114122 T^{2} + p^{8} T^{4}$$
19$C_2^2$ $$1 - 250922 T^{2} + p^{8} T^{4}$$
23$C_2$ $$( 1 - 277 T + p^{4} T^{2} )^{2}$$
29$C_2$ $$( 1 - 38 p T + p^{4} T^{2} )( 1 + 38 p T + p^{4} T^{2} )$$
31$C_2$ $$( 1 + 1363 T + p^{4} T^{2} )^{2}$$
37$C_2$ $$( 1 - 167 T + p^{4} T^{2} )^{2}$$
41$C_2^2$ $$1 - 4522442 T^{2} + p^{8} T^{4}$$
43$C_2^2$ $$1 - 5385602 T^{2} + p^{8} T^{4}$$
47$C_2$ $$( 1 - 1702 T + p^{4} T^{2} )^{2}$$
53$C_2$ $$( 1 - 4522 T + p^{4} T^{2} )^{2}$$
59$C_2$ $$( 1 + 2363 T + p^{4} T^{2} )^{2}$$
61$C_2^2$ $$1 - 11966402 T^{2} + p^{8} T^{4}$$
67$C_2$ $$( 1 + 2803 T + p^{4} T^{2} )^{2}$$
71$C_2$ $$( 1 - 3397 T + p^{4} T^{2} )^{2}$$
73$C_2^2$ $$1 - 45779402 T^{2} + p^{8} T^{4}$$
79$C_2^2$ $$1 - 40803842 T^{2} + p^{8} T^{4}$$
83$C_2^2$ $$1 - 94223522 T^{2} + p^{8} T^{4}$$
89$C_2$ $$( 1 + 4673 T + p^{4} T^{2} )^{2}$$
97$C_2$ $$( 1 - 4247 T + p^{4} T^{2} )^{2}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}