# Properties

 Label 4-2352e2-1.1-c3e2-0-16 Degree $4$ Conductor $5531904$ Sign $1$ Analytic cond. $19257.8$ Root an. cond. $11.7801$ Motivic weight $3$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $2$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 6·3-s − 5·5-s + 27·9-s − 67·11-s − 41·13-s − 30·15-s + 92·17-s + 43·19-s − 148·23-s + 105·25-s + 108·27-s + 77·29-s − 520·31-s − 402·33-s + 7·37-s − 246·39-s + 426·41-s + 107·43-s − 135·45-s + 576·47-s + 552·51-s − 243·53-s + 335·55-s + 258·57-s − 7·59-s − 224·61-s + 205·65-s + ⋯
 L(s)  = 1 + 1.15·3-s − 0.447·5-s + 9-s − 1.83·11-s − 0.874·13-s − 0.516·15-s + 1.31·17-s + 0.519·19-s − 1.34·23-s + 0.839·25-s + 0.769·27-s + 0.493·29-s − 3.01·31-s − 2.12·33-s + 0.0311·37-s − 1.01·39-s + 1.62·41-s + 0.379·43-s − 0.447·45-s + 1.78·47-s + 1.51·51-s − 0.629·53-s + 0.821·55-s + 0.599·57-s − 0.0154·59-s − 0.470·61-s + 0.391·65-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 5531904 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 5531904 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$4$$ Conductor: $$5531904$$    =    $$2^{8} \cdot 3^{2} \cdot 7^{4}$$ Sign: $1$ Analytic conductor: $$19257.8$$ Root analytic conductor: $$11.7801$$ Motivic weight: $$3$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{2352} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$2$$ Selberg data: $$(4,\ 5531904,\ (\ :3/2, 3/2),\ 1)$$

## 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)$
bad2 $$1$$
3$C_1$ $$( 1 - p T )^{2}$$
7 $$1$$
good5$D_{4}$ $$1 + p T - 16 p T^{2} + p^{4} T^{3} + p^{6} T^{4}$$
11$D_{4}$ $$1 + 67 T + 3448 T^{2} + 67 p^{3} T^{3} + p^{6} T^{4}$$
13$D_{4}$ $$1 + 41 T + 4478 T^{2} + 41 p^{3} T^{3} + p^{6} T^{4}$$
17$D_{4}$ $$1 - 92 T + 386 p T^{2} - 92 p^{3} T^{3} + p^{6} T^{4}$$
19$D_{4}$ $$1 - 43 T + 11154 T^{2} - 43 p^{3} T^{3} + p^{6} T^{4}$$
23$D_{4}$ $$1 + 148 T + 24430 T^{2} + 148 p^{3} T^{3} + p^{6} T^{4}$$
29$D_{4}$ $$1 - 77 T + 9574 T^{2} - 77 p^{3} T^{3} + p^{6} T^{4}$$
31$D_{4}$ $$1 + 520 T + 125837 T^{2} + 520 p^{3} T^{3} + p^{6} T^{4}$$
37$D_{4}$ $$1 - 7 T + 74082 T^{2} - 7 p^{3} T^{3} + p^{6} T^{4}$$
41$D_{4}$ $$1 - 426 T + 171106 T^{2} - 426 p^{3} T^{3} + p^{6} T^{4}$$
43$D_{4}$ $$1 - 107 T + 86220 T^{2} - 107 p^{3} T^{3} + p^{6} T^{4}$$
47$D_{4}$ $$1 - 576 T + 242170 T^{2} - 576 p^{3} T^{3} + p^{6} T^{4}$$
53$D_{4}$ $$1 + 243 T + 309490 T^{2} + 243 p^{3} T^{3} + p^{6} T^{4}$$
59$D_{4}$ $$1 + 7 T + 200614 T^{2} + 7 p^{3} T^{3} + p^{6} T^{4}$$
61$D_{4}$ $$1 + 224 T + 461126 T^{2} + 224 p^{3} T^{3} + p^{6} T^{4}$$
67$D_{4}$ $$1 + 687 T + 678832 T^{2} + 687 p^{3} T^{3} + p^{6} T^{4}$$
71$D_{4}$ $$1 + 472 T + 637018 T^{2} + 472 p^{3} T^{3} + p^{6} T^{4}$$
73$D_{4}$ $$1 - 921 T + 578188 T^{2} - 921 p^{3} T^{3} + p^{6} T^{4}$$
79$D_{4}$ $$1 - 526 T + 1033727 T^{2} - 526 p^{3} T^{3} + p^{6} T^{4}$$
83$D_{4}$ $$1 + 221 T + 945628 T^{2} + 221 p^{3} T^{3} + p^{6} T^{4}$$
89$D_{4}$ $$1 + 774 T + 966562 T^{2} + 774 p^{3} T^{3} + p^{6} T^{4}$$
97$D_{4}$ $$1 + 1953 T + 2366992 T^{2} + 1953 p^{3} T^{3} + p^{6} T^{4}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$