# Properties

 Label 4-103e2-1.1-c1e2-0-0 Degree $4$ Conductor $10609$ Sign $-1$ Analytic cond. $0.676439$ Root an. cond. $0.906895$ Motivic weight $1$ Arithmetic yes Rational yes Primitive yes Self-dual yes Analytic rank $1$

# Origins

## Dirichlet series

 L(s)  = 1 − 2-s − 3-s + 5-s + 6-s − 2·7-s − 8-s − 2·9-s − 10-s − 3·13-s + 2·14-s − 15-s − 16-s − 4·17-s + 2·18-s − 3·19-s + 2·21-s − 2·23-s + 24-s + 4·25-s + 3·26-s + 2·27-s + 29-s + 30-s − 2·31-s + 6·32-s + 4·34-s − 2·35-s + ⋯
 L(s)  = 1 − 0.707·2-s − 0.577·3-s + 0.447·5-s + 0.408·6-s − 0.755·7-s − 0.353·8-s − 2/3·9-s − 0.316·10-s − 0.832·13-s + 0.534·14-s − 0.258·15-s − 1/4·16-s − 0.970·17-s + 0.471·18-s − 0.688·19-s + 0.436·21-s − 0.417·23-s + 0.204·24-s + 4/5·25-s + 0.588·26-s + 0.384·27-s + 0.185·29-s + 0.182·30-s − 0.359·31-s + 1.06·32-s + 0.685·34-s − 0.338·35-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 10609 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 10609 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$4$$ Conductor: $$10609$$    =    $$103^{2}$$ Sign: $-1$ Analytic conductor: $$0.676439$$ Root analytic conductor: $$0.906895$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: yes Self-dual: yes Analytic rank: $$1$$ Selberg data: $$(4,\ 10609,\ (\ :1/2, 1/2),\ -1)$$

## Particular Values

 $$L(1)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{3}{2})$$ 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)$
bad103$C_2$ $$1 - 11 T + p T^{2}$$
good2$D_{4}$ $$1 + T + T^{2} + p T^{3} + p^{2} T^{4}$$
3$D_{4}$ $$1 + T + p T^{2} + p T^{3} + p^{2} T^{4}$$
5$D_{4}$ $$1 - T - 3 T^{2} - p T^{3} + p^{2} T^{4}$$
7$C_2$$\times$$C_2$ $$( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} )$$
11$C_2^2$ $$1 - 10 T^{2} + p^{2} T^{4}$$
13$D_{4}$ $$1 + 3 T + 27 T^{2} + 3 p T^{3} + p^{2} T^{4}$$
17$D_{4}$ $$1 + 4 T + 25 T^{2} + 4 p T^{3} + p^{2} T^{4}$$
19$D_{4}$ $$1 + 3 T + 17 T^{2} + 3 p T^{3} + p^{2} T^{4}$$
23$D_{4}$ $$1 + 2 T + 32 T^{2} + 2 p T^{3} + p^{2} T^{4}$$
29$D_{4}$ $$1 - T - 11 T^{2} - p T^{3} + p^{2} T^{4}$$
31$D_{4}$ $$1 + 2 T + 34 T^{2} + 2 p T^{3} + p^{2} T^{4}$$
37$D_{4}$ $$1 - 7 T + 35 T^{2} - 7 p T^{3} + p^{2} T^{4}$$
41$D_{4}$ $$1 + 4 T - 22 T^{2} + 4 p T^{3} + p^{2} T^{4}$$
43$D_{4}$ $$1 + 3 T + 25 T^{2} + 3 p T^{3} + p^{2} T^{4}$$
47$D_{4}$ $$1 + T + 79 T^{2} + p T^{3} + p^{2} T^{4}$$
53$D_{4}$ $$1 + 7 T + 33 T^{2} + 7 p T^{3} + p^{2} T^{4}$$
59$D_{4}$ $$1 - 5 T + 67 T^{2} - 5 p T^{3} + p^{2} T^{4}$$
61$D_{4}$ $$1 + 8 T + 90 T^{2} + 8 p T^{3} + p^{2} T^{4}$$
67$D_{4}$ $$1 - 7 T + 139 T^{2} - 7 p T^{3} + p^{2} T^{4}$$
71$D_{4}$ $$1 + 6 T + 67 T^{2} + 6 p T^{3} + p^{2} T^{4}$$
73$C_2^2$ $$1 + 14 T + 123 T^{2} + 14 p T^{3} + p^{2} T^{4}$$
79$D_{4}$ $$1 + T - 87 T^{2} + p T^{3} + p^{2} T^{4}$$
83$D_{4}$ $$1 - 14 T + 128 T^{2} - 14 p T^{3} + p^{2} T^{4}$$
89$D_{4}$ $$1 + 2 T + 146 T^{2} + 2 p T^{3} + p^{2} T^{4}$$
97$D_{4}$ $$1 - 10 T + 176 T^{2} - 10 p T^{3} + p^{2} T^{4}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$