# Properties

 Label 4-112e2-1.1-c13e2-0-1 Degree $4$ Conductor $12544$ Sign $1$ Analytic cond. $14423.6$ Root an. cond. $10.9589$ Motivic weight $13$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $2$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 1.10e3·3-s + 7.55e4·5-s + 2.35e5·7-s − 2.96e5·9-s − 3.33e6·11-s + 7.99e6·13-s − 8.35e7·15-s + 3.90e7·17-s − 1.24e8·19-s − 2.60e8·21-s − 1.90e9·23-s + 1.95e9·25-s + 2.45e8·27-s − 2.45e8·29-s + 1.10e10·31-s + 3.68e9·33-s + 1.77e10·35-s − 3.96e10·37-s − 8.84e9·39-s − 2.43e9·41-s − 1.78e10·43-s − 2.24e10·45-s − 6.44e10·47-s + 4.15e10·49-s − 4.31e10·51-s − 1.26e11·53-s − 2.51e11·55-s + ⋯
 L(s)  = 1 − 0.875·3-s + 2.16·5-s + 0.755·7-s − 0.186·9-s − 0.567·11-s + 0.459·13-s − 1.89·15-s + 0.392·17-s − 0.605·19-s − 0.662·21-s − 2.67·23-s + 1.60·25-s + 0.122·27-s − 0.0765·29-s + 2.22·31-s + 0.497·33-s + 1.63·35-s − 2.53·37-s − 0.402·39-s − 0.0799·41-s − 0.429·43-s − 0.402·45-s − 0.872·47-s + 3/7·49-s − 0.343·51-s − 0.783·53-s − 1.22·55-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 12544 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(14-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 12544 ^{s/2} \, \Gamma_{\C}(s+13/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$4$$ Conductor: $$12544$$    =    $$2^{8} \cdot 7^{2}$$ Sign: $1$ Analytic conductor: $$14423.6$$ Root analytic conductor: $$10.9589$$ Motivic weight: $$13$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{112} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$2$$ Selberg data: $$(4,\ 12544,\ (\ :13/2, 13/2),\ 1)$$

## Particular Values

 $$L(7)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{15}{2})$$ 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)$
bad2 $$1$$
7$C_1$ $$( 1 - p^{6} T )^{2}$$
good3$D_{4}$ $$1 + 1106 T + 56290 p^{3} T^{2} + 1106 p^{13} T^{3} + p^{26} T^{4}$$
5$D_{4}$ $$1 - 15106 p T + 749493058 p T^{2} - 15106 p^{14} T^{3} + p^{26} T^{4}$$
11$D_{4}$ $$1 + 303260 p T - 17283060398 p^{3} T^{2} + 303260 p^{14} T^{3} + p^{26} T^{4}$$
13$D_{4}$ $$1 - 7998102 T + 313454244594482 T^{2} - 7998102 p^{13} T^{3} + p^{26} T^{4}$$
17$D_{4}$ $$1 - 39024832 T + 10050449815930430 T^{2} - 39024832 p^{13} T^{3} + p^{26} T^{4}$$
19$D_{4}$ $$1 + 124092934 T + 64593685044636582 T^{2} + 124092934 p^{13} T^{3} + p^{26} T^{4}$$
23$D_{4}$ $$1 + 1900816000 T + 1895455828613638766 T^{2} + 1900816000 p^{13} T^{3} + p^{26} T^{4}$$
29$D_{4}$ $$1 + 245135152 T + 11525694953905661654 T^{2} + 245135152 p^{13} T^{3} + p^{26} T^{4}$$
31$D_{4}$ $$1 - 11010560148 T + 72353550492368544158 T^{2} - 11010560148 p^{13} T^{3} + p^{26} T^{4}$$
37$D_{4}$ $$1 + 39630491216 T +$$$$77\!\cdots\!58$$$$T^{2} + 39630491216 p^{13} T^{3} + p^{26} T^{4}$$
41$D_{4}$ $$1 + 2431027368 T +$$$$18\!\cdots\!98$$$$T^{2} + 2431027368 p^{13} T^{3} + p^{26} T^{4}$$
43$D_{4}$ $$1 + 17823138316 T +$$$$10\!\cdots\!50$$$$T^{2} + 17823138316 p^{13} T^{3} + p^{26} T^{4}$$
47$D_{4}$ $$1 + 64488311076 T +$$$$22\!\cdots\!98$$$$T^{2} + 64488311076 p^{13} T^{3} + p^{26} T^{4}$$
53$D_{4}$ $$1 + 126504176628 T +$$$$48\!\cdots\!42$$$$T^{2} + 126504176628 p^{13} T^{3} + p^{26} T^{4}$$
59$D_{4}$ $$1 + 341259961238 T +$$$$17\!\cdots\!94$$$$T^{2} + 341259961238 p^{13} T^{3} + p^{26} T^{4}$$
61$D_{4}$ $$1 - 447240700746 T +$$$$34\!\cdots\!66$$$$T^{2} - 447240700746 p^{13} T^{3} + p^{26} T^{4}$$
67$D_{4}$ $$1 - 2071322290888 T +$$$$21\!\cdots\!10$$$$T^{2} - 2071322290888 p^{13} T^{3} + p^{26} T^{4}$$
71$D_{4}$ $$1 + 650434465720 T +$$$$22\!\cdots\!22$$$$T^{2} + 650434465720 p^{13} T^{3} + p^{26} T^{4}$$
73$D_{4}$ $$1 + 1449809330116 T +$$$$32\!\cdots\!30$$$$T^{2} + 1449809330116 p^{13} T^{3} + p^{26} T^{4}$$
79$D_{4}$ $$1 + 1525152397656 T +$$$$78\!\cdots\!62$$$$T^{2} + 1525152397656 p^{13} T^{3} + p^{26} T^{4}$$
83$D_{4}$ $$1 + 4257517639438 T +$$$$13\!\cdots\!62$$$$T^{2} + 4257517639438 p^{13} T^{3} + p^{26} T^{4}$$
89$D_{4}$ $$1 - 12593651222628 T +$$$$83\!\cdots\!34$$$$T^{2} - 12593651222628 p^{13} T^{3} + p^{26} T^{4}$$
97$D_{4}$ $$1 + 16541570007760 T +$$$$16\!\cdots\!54$$$$T^{2} + 16541570007760 p^{13} T^{3} + p^{26} T^{4}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$