# Properties

 Label 4-7e4-1.1-c5e2-0-3 Degree $4$ Conductor $2401$ Sign $1$ Analytic cond. $61.7608$ Root an. cond. $2.80335$ Motivic weight $5$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 2·2-s + 8·3-s − 24·4-s + 38·5-s + 16·6-s − 40·8-s − 401·9-s + 76·10-s + 424·11-s − 192·12-s + 924·13-s + 304·15-s − 304·16-s + 2.34e3·17-s − 802·18-s + 360·19-s − 912·20-s + 848·22-s − 12·23-s − 320·24-s − 1.46e3·25-s + 1.84e3·26-s − 4.98e3·27-s − 7.05e3·29-s + 608·30-s − 3.54e3·31-s − 3.07e3·32-s + ⋯
 L(s)  = 1 + 0.353·2-s + 0.513·3-s − 3/4·4-s + 0.679·5-s + 0.181·6-s − 0.220·8-s − 1.65·9-s + 0.240·10-s + 1.05·11-s − 0.384·12-s + 1.51·13-s + 0.348·15-s − 0.296·16-s + 1.96·17-s − 0.583·18-s + 0.228·19-s − 0.509·20-s + 0.373·22-s − 0.00473·23-s − 0.113·24-s − 0.469·25-s + 0.536·26-s − 1.31·27-s − 1.55·29-s + 0.123·30-s − 0.663·31-s − 0.530·32-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 2401 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 2401 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$4$$ Conductor: $$2401$$    =    $$7^{4}$$ Sign: $1$ Analytic conductor: $$61.7608$$ Root analytic conductor: $$2.80335$$ Motivic weight: $$5$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(4,\ 2401,\ (\ :5/2, 5/2),\ 1)$$

## Particular Values

 $$L(3)$$ $$\approx$$ $$2.863849664$$ $$L(\frac12)$$ $$\approx$$ $$2.863849664$$ $$L(\frac{7}{2})$$ 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)$
bad7 $$1$$
good2$D_{4}$ $$1 - p T + 7 p^{2} T^{2} - p^{6} T^{3} + p^{10} T^{4}$$
3$D_{4}$ $$1 - 8 T + 155 p T^{2} - 8 p^{5} T^{3} + p^{10} T^{4}$$
5$D_{4}$ $$1 - 38 T + 2911 T^{2} - 38 p^{5} T^{3} + p^{10} T^{4}$$
11$D_{4}$ $$1 - 424 T + 347473 T^{2} - 424 p^{5} T^{3} + p^{10} T^{4}$$
13$D_{4}$ $$1 - 924 T + 927022 T^{2} - 924 p^{5} T^{3} + p^{10} T^{4}$$
17$D_{4}$ $$1 - 138 p T + 3570955 T^{2} - 138 p^{6} T^{3} + p^{10} T^{4}$$
19$D_{4}$ $$1 - 360 T + 2145625 T^{2} - 360 p^{5} T^{3} + p^{10} T^{4}$$
23$D_{4}$ $$1 + 12 T + 12696565 T^{2} + 12 p^{5} T^{3} + p^{10} T^{4}$$
29$D_{4}$ $$1 + 7052 T + 35324974 T^{2} + 7052 p^{5} T^{3} + p^{10} T^{4}$$
31$D_{4}$ $$1 + 3548 T + 41490053 T^{2} + 3548 p^{5} T^{3} + p^{10} T^{4}$$
37$D_{4}$ $$1 - 11090 T + 146343239 T^{2} - 11090 p^{5} T^{3} + p^{10} T^{4}$$
41$D_{4}$ $$1 + 3500 T + 206898214 T^{2} + 3500 p^{5} T^{3} + p^{10} T^{4}$$
43$D_{4}$ $$1 + 12680 T + 267378054 T^{2} + 12680 p^{5} T^{3} + p^{10} T^{4}$$
47$D_{4}$ $$1 - 22956 T + 491525173 T^{2} - 22956 p^{5} T^{3} + p^{10} T^{4}$$
53$D_{4}$ $$1 - 3042 T + 716414839 T^{2} - 3042 p^{5} T^{3} + p^{10} T^{4}$$
59$D_{4}$ $$1 - 65808 T + 2502089257 T^{2} - 65808 p^{5} T^{3} + p^{10} T^{4}$$
61$D_{4}$ $$1 - 42486 T + 1501996159 T^{2} - 42486 p^{5} T^{3} + p^{10} T^{4}$$
67$D_{4}$ $$1 - 42312 T + 3116577793 T^{2} - 42312 p^{5} T^{3} + p^{10} T^{4}$$
71$D_{4}$ $$1 + 2208 T + 3433192846 T^{2} + 2208 p^{5} T^{3} + p^{10} T^{4}$$
73$D_{4}$ $$1 - 50506 T + 2773040987 T^{2} - 50506 p^{5} T^{3} + p^{10} T^{4}$$
79$D_{4}$ $$1 - 9004 T + 5176592589 T^{2} - 9004 p^{5} T^{3} + p^{10} T^{4}$$
83$D_{4}$ $$1 - 104328 T + 9837878230 T^{2} - 104328 p^{5} T^{3} + p^{10} T^{4}$$
89$D_{4}$ $$1 - 26666 T + 8107102555 T^{2} - 26666 p^{5} T^{3} + p^{10} T^{4}$$
97$D_{4}$ $$1 + 2156 p T + 28107307478 T^{2} + 2156 p^{6} T^{3} + p^{10} T^{4}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$