# Properties

 Label 2-1560-1560.389-c0-0-4 Degree $2$ Conductor $1560$ Sign $1$ Analytic cond. $0.778541$ Root an. cond. $0.882349$ Motivic weight $0$ Arithmetic yes Rational yes Primitive yes Self-dual yes Analytic rank $0$

# Origins

## Dirichlet series

 L(s)  = 1 − 2-s + 3-s + 4-s + 5-s − 6-s − 8-s + 9-s − 10-s + 12-s − 13-s + 15-s + 16-s − 18-s + 20-s − 24-s + 25-s + 26-s + 27-s − 30-s − 32-s + 36-s − 39-s − 40-s + 2·41-s − 2·43-s + 45-s + 48-s + ⋯
 L(s)  = 1 − 2-s + 3-s + 4-s + 5-s − 6-s − 8-s + 9-s − 10-s + 12-s − 13-s + 15-s + 16-s − 18-s + 20-s − 24-s + 25-s + 26-s + 27-s − 30-s − 32-s + 36-s − 39-s − 40-s + 2·41-s − 2·43-s + 45-s + 48-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$1560$$    =    $$2^{3} \cdot 3 \cdot 5 \cdot 13$$ Sign: $1$ Analytic conductor: $$0.778541$$ Root analytic conductor: $$0.882349$$ Motivic weight: $$0$$ Rational: yes Arithmetic: yes Character: $\chi_{1560} (389, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(2,\ 1560,\ (\ :0),\ 1)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$1.198582535$$ $$L(\frac12)$$ $$\approx$$ $$1.198582535$$ $$L(1)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + T$$
3 $$1 - T$$
5 $$1 - T$$
13 $$1 + T$$
good7 $$1 + T^{2}$$
11 $$( 1 - T )( 1 + T )$$
17 $$1 + T^{2}$$
19 $$1 + T^{2}$$
23 $$1 + T^{2}$$
29 $$1 + T^{2}$$
31 $$( 1 - T )( 1 + T )$$
37 $$( 1 - T )( 1 + T )$$
41 $$( 1 - T )^{2}$$
43 $$( 1 + T )^{2}$$
47 $$( 1 - T )( 1 + T )$$
53 $$( 1 - T )( 1 + T )$$
59 $$( 1 - T )( 1 + T )$$
61 $$( 1 - T )( 1 + T )$$
67 $$( 1 - T )( 1 + T )$$
71 $$( 1 + T )^{2}$$
73 $$1 + T^{2}$$
79 $$( 1 + T )^{2}$$
83 $$( 1 - T )^{2}$$
89 $$( 1 + T )^{2}$$
97 $$1 + T^{2}$$
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$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$

## Imaginary part of the first few zeros on the critical line

−9.617491577200472130034388989993, −8.941424809775066615634410648681, −8.194792527080159563460027048944, −7.37310148028000106128810082902, −6.71614466108682143295487482295, −5.74993956813084240037296754846, −4.61456148074193800106382257507, −3.16313911805192908399185715721, −2.40660888618246790221497614286, −1.50135032922634249265141050625, 1.50135032922634249265141050625, 2.40660888618246790221497614286, 3.16313911805192908399185715721, 4.61456148074193800106382257507, 5.74993956813084240037296754846, 6.71614466108682143295487482295, 7.37310148028000106128810082902, 8.194792527080159563460027048944, 8.941424809775066615634410648681, 9.617491577200472130034388989993