# Properties

 Label 4-13e2-1.1-c3e2-0-3 Degree $4$ Conductor $169$ Sign $1$ Analytic cond. $0.588327$ Root an. cond. $0.875799$ Motivic weight $3$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 2-s + 5·3-s − 11·4-s − 3·5-s + 5·6-s − 9·7-s − 15·8-s + 3·9-s − 3·10-s + 80·11-s − 55·12-s − 26·13-s − 9·14-s − 15·15-s + 61·16-s + 19·17-s + 3·18-s − 84·19-s + 33·20-s − 45·21-s + 80·22-s + 196·23-s − 75·24-s − 239·25-s − 26·26-s + 40·27-s + 99·28-s + ⋯
 L(s)  = 1 + 0.353·2-s + 0.962·3-s − 1.37·4-s − 0.268·5-s + 0.340·6-s − 0.485·7-s − 0.662·8-s + 1/9·9-s − 0.0948·10-s + 2.19·11-s − 1.32·12-s − 0.554·13-s − 0.171·14-s − 0.258·15-s + 0.953·16-s + 0.271·17-s + 0.0392·18-s − 1.01·19-s + 0.368·20-s − 0.467·21-s + 0.775·22-s + 1.77·23-s − 0.637·24-s − 1.91·25-s − 0.196·26-s + 0.285·27-s + 0.668·28-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$4$$ Conductor: $$169$$    =    $$13^{2}$$ Sign: $1$ Analytic conductor: $$0.588327$$ Root analytic conductor: $$0.875799$$ Motivic weight: $$3$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(4,\ 169,\ (\ :3/2, 3/2),\ 1)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$1.009426291$$ $$L(\frac12)$$ $$\approx$$ $$1.009426291$$ $$L(\frac{5}{2})$$ 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$C_1$ $$( 1 + p T )^{2}$$
good2$D_{4}$ $$1 - T + 3 p^{2} T^{2} - p^{3} T^{3} + p^{6} T^{4}$$
3$D_{4}$ $$1 - 5 T + 22 T^{2} - 5 p^{3} T^{3} + p^{6} T^{4}$$
5$D_{4}$ $$1 + 3 T + 248 T^{2} + 3 p^{3} T^{3} + p^{6} T^{4}$$
7$D_{4}$ $$1 + 9 T + 192 T^{2} + 9 p^{3} T^{3} + p^{6} T^{4}$$
11$D_{4}$ $$1 - 80 T + 3650 T^{2} - 80 p^{3} T^{3} + p^{6} T^{4}$$
17$D_{4}$ $$1 - 19 T + 8688 T^{2} - 19 p^{3} T^{3} + p^{6} T^{4}$$
19$D_{4}$ $$1 + 84 T + 11130 T^{2} + 84 p^{3} T^{3} + p^{6} T^{4}$$
23$D_{4}$ $$1 - 196 T + 33326 T^{2} - 196 p^{3} T^{3} + p^{6} T^{4}$$
29$D_{4}$ $$1 + 44 T + 10094 T^{2} + 44 p^{3} T^{3} + p^{6} T^{4}$$
31$D_{4}$ $$1 + 86 T + 56518 T^{2} + 86 p^{3} T^{3} + p^{6} T^{4}$$
37$D_{4}$ $$1 - 209 T + 112120 T^{2} - 209 p^{3} T^{3} + p^{6} T^{4}$$
41$D_{4}$ $$1 + 230 T + 149010 T^{2} + 230 p^{3} T^{3} + p^{6} T^{4}$$
43$D_{4}$ $$1 - 287 T + 92698 T^{2} - 287 p^{3} T^{3} + p^{6} T^{4}$$
47$D_{4}$ $$1 - 435 T + 192728 T^{2} - 435 p^{3} T^{3} + p^{6} T^{4}$$
53$D_{4}$ $$1 + 118 T + 297410 T^{2} + 118 p^{3} T^{3} + p^{6} T^{4}$$
59$D_{4}$ $$1 + 368 T + 379266 T^{2} + 368 p^{3} T^{3} + p^{6} T^{4}$$
61$D_{4}$ $$1 + 1058 T + 580378 T^{2} + 1058 p^{3} T^{3} + p^{6} T^{4}$$
67$D_{4}$ $$1 - 68 T + 373930 T^{2} - 68 p^{3} T^{3} + p^{6} T^{4}$$
71$D_{4}$ $$1 + 131 T + 493328 T^{2} + 131 p^{3} T^{3} + p^{6} T^{4}$$
73$D_{4}$ $$1 - 456 T + 542718 T^{2} - 456 p^{3} T^{3} + p^{6} T^{4}$$
79$D_{4}$ $$1 + 1008 T + 1233294 T^{2} + 1008 p^{3} T^{3} + p^{6} T^{4}$$
83$D_{4}$ $$1 - 1958 T + 1961238 T^{2} - 1958 p^{3} T^{3} + p^{6} T^{4}$$
89$D_{4}$ $$1 + 720 T + 899726 T^{2} + 720 p^{3} T^{3} + p^{6} T^{4}$$
97$D_{4}$ $$1 + 928 T + 943870 T^{2} + 928 p^{3} T^{3} + p^{6} T^{4}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$