# Properties

 Label 2-2240-35.34-c0-0-6 Degree $2$ Conductor $2240$ Sign $1$ Analytic cond. $1.11790$ Root an. cond. $1.05731$ Motivic weight $0$ Arithmetic yes Rational yes Primitive yes Self-dual yes Analytic rank $0$

# Origins

## Dirichlet series

 L(s)  = 1 + 3-s + 5-s + 7-s − 11-s − 13-s + 15-s + 17-s + 21-s + 25-s − 27-s + 29-s − 33-s + 35-s − 39-s − 47-s + 49-s + 51-s − 55-s − 65-s − 2·71-s − 2·73-s + 75-s − 77-s + 79-s − 81-s − 2·83-s + 85-s + ⋯
 L(s)  = 1 + 3-s + 5-s + 7-s − 11-s − 13-s + 15-s + 17-s + 21-s + 25-s − 27-s + 29-s − 33-s + 35-s − 39-s − 47-s + 49-s + 51-s − 55-s − 65-s − 2·71-s − 2·73-s + 75-s − 77-s + 79-s − 81-s − 2·83-s + 85-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$2240$$    =    $$2^{6} \cdot 5 \cdot 7$$ Sign: $1$ Analytic conductor: $$1.11790$$ Root analytic conductor: $$1.05731$$ Motivic weight: $$0$$ Rational: yes Arithmetic: yes Character: $\chi_{2240} (769, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(2,\ 2240,\ (\ :0),\ 1)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$1.958626298$$ $$L(\frac12)$$ $$\approx$$ $$1.958626298$$ $$L(1)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
5 $$1 - T$$
7 $$1 - T$$
good3 $$1 - T + T^{2}$$
11 $$1 + T + T^{2}$$
13 $$1 + T + T^{2}$$
17 $$1 - T + T^{2}$$
19 $$( 1 - T )( 1 + T )$$
23 $$( 1 - T )( 1 + T )$$
29 $$1 - T + T^{2}$$
31 $$( 1 - T )( 1 + T )$$
37 $$( 1 - T )( 1 + T )$$
41 $$( 1 - T )( 1 + T )$$
43 $$( 1 - T )( 1 + T )$$
47 $$1 + T + T^{2}$$
53 $$( 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 + T^{2}$$
83 $$( 1 + T )^{2}$$
89 $$( 1 - T )( 1 + T )$$
97 $$1 - T + 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

−9.124509353505063248333293657547, −8.437391199479060024941774546356, −7.81843726316528054009667193924, −7.17044445879043346687486974805, −5.91316605065441977695989779646, −5.24734081919091668582870628636, −4.52378910882817524591378354822, −3.07890357616003525241526909737, −2.51807976290935731715877354947, −1.57296568307498081096017582090, 1.57296568307498081096017582090, 2.51807976290935731715877354947, 3.07890357616003525241526909737, 4.52378910882817524591378354822, 5.24734081919091668582870628636, 5.91316605065441977695989779646, 7.17044445879043346687486974805, 7.81843726316528054009667193924, 8.437391199479060024941774546356, 9.124509353505063248333293657547