# Properties

 Label 4-13e4-1.1-c3e2-0-13 Degree $4$ Conductor $28561$ Sign $1$ Analytic cond. $99.4272$ Root an. cond. $3.15774$ Motivic weight $3$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $2$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 14·3-s − 4·4-s + 93·9-s + 56·12-s − 48·16-s − 54·17-s − 114·23-s − 58·25-s − 238·27-s − 138·29-s − 372·36-s + 170·43-s + 672·48-s − 179·49-s + 756·51-s + 852·53-s − 34·61-s + 448·64-s + 216·68-s + 1.59e3·69-s + 812·75-s − 2.48e3·79-s − 1.68e3·81-s + 1.93e3·87-s + 456·92-s + 232·100-s − 3.91e3·101-s + ⋯
 L(s)  = 1 − 2.69·3-s − 1/2·4-s + 31/9·9-s + 1.34·12-s − 3/4·16-s − 0.770·17-s − 1.03·23-s − 0.463·25-s − 1.69·27-s − 0.883·29-s − 1.72·36-s + 0.602·43-s + 2.02·48-s − 0.521·49-s + 2.07·51-s + 2.20·53-s − 0.0713·61-s + 7/8·64-s + 0.385·68-s + 2.78·69-s + 1.25·75-s − 3.54·79-s − 2.31·81-s + 2.38·87-s + 0.516·92-s + 0.231·100-s − 3.85·101-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 28561 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 28561 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$4$$ Conductor: $$28561$$    =    $$13^{4}$$ Sign: $1$ Analytic conductor: $$99.4272$$ Root analytic conductor: $$3.15774$$ Motivic weight: $$3$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{169} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$2$$ Selberg data: $$(4,\ 28561,\ (\ :3/2, 3/2),\ 1)$$

## Particular Values

 $$L(2)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{5}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$\Gal(F_p)$$F_p(T)$
bad13 $$1$$
good2$C_2^2$ $$1 + p^{2} T^{2} + p^{6} T^{4}$$
3$C_2$ $$( 1 + 7 T + p^{3} T^{2} )^{2}$$
5$C_2^2$ $$1 + 58 T^{2} + p^{6} T^{4}$$
7$C_2^2$ $$1 + 179 T^{2} + p^{6} T^{4}$$
11$C_2^2$ $$1 + 2155 T^{2} + p^{6} T^{4}$$
17$C_2$ $$( 1 + 27 T + p^{3} T^{2} )^{2}$$
19$C_2^2$ $$1 + 5915 T^{2} + p^{6} T^{4}$$
23$C_2$ $$( 1 + 57 T + p^{3} T^{2} )^{2}$$
29$C_2$ $$( 1 + 69 T + p^{3} T^{2} )^{2}$$
31$C_2^2$ $$1 + 54290 T^{2} + p^{6} T^{4}$$
37$C_2^2$ $$1 + 99719 T^{2} + p^{6} T^{4}$$
41$C_2^2$ $$1 - 16745 T^{2} + p^{6} T^{4}$$
43$C_2$ $$( 1 - 85 T + p^{3} T^{2} )^{2}$$
47$C_2^2$ $$1 + 90034 T^{2} + p^{6} T^{4}$$
53$C_2$ $$( 1 - 426 T + p^{3} T^{2} )^{2}$$
59$C_2^2$ $$1 + 410395 T^{2} + p^{6} T^{4}$$
61$C_2$ $$( 1 + 17 T + p^{3} T^{2} )^{2}$$
67$C_2^2$ $$1 + 574451 T^{2} + p^{6} T^{4}$$
71$C_2^2$ $$1 + 375115 T^{2} + p^{6} T^{4}$$
73$C_2^2$ $$1 - 231166 T^{2} + p^{6} T^{4}$$
79$C_2$ $$( 1 + 1244 T + p^{3} T^{2} )^{2}$$
83$C_2^2$ $$1 + 962026 T^{2} + p^{6} T^{4}$$
89$C_2^2$ $$1 + 1315951 T^{2} + p^{6} T^{4}$$
97$C_2^2$ $$1 + 300239 T^{2} + p^{6} T^{4}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$