# Properties

 Degree $2$ Conductor $1575$ Sign $0.894 + 0.447i$ Motivic weight $0$ Primitive yes Self-dual no Analytic rank $0$

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## Dirichlet series

 L(s)  = 1 + 4-s − i·7-s + 16-s − i·28-s − 2i·37-s + 2i·43-s − 49-s + 64-s + 2i·67-s − 2·79-s + 2·109-s − i·112-s + ⋯
 L(s)  = 1 + 4-s − i·7-s + 16-s − i·28-s − 2i·37-s + 2i·43-s − 49-s + 64-s + 2i·67-s − 2·79-s + 2·109-s − i·112-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$1575$$    =    $$3^{2} \cdot 5^{2} \cdot 7$$ Sign: $0.894 + 0.447i$ Motivic weight: $$0$$ Character: $\chi_{1575} (874, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 1575,\ (\ :0),\ 0.894 + 0.447i)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$1.405897070$$ $$L(\frac12)$$ $$\approx$$ $$1.405897070$$ $$L(1)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1$$
5 $$1$$
7 $$1 + iT$$
good2 $$1 - T^{2}$$
11 $$1 + T^{2}$$
13 $$1 + T^{2}$$
17 $$1 + T^{2}$$
19 $$1 - T^{2}$$
23 $$1 - T^{2}$$
29 $$1 + T^{2}$$
31 $$1 - T^{2}$$
37 $$1 + 2iT - T^{2}$$
41 $$1 - T^{2}$$
43 $$1 - 2iT - T^{2}$$
47 $$1 + T^{2}$$
53 $$1 - T^{2}$$
59 $$1 - T^{2}$$
61 $$1 - T^{2}$$
67 $$1 - 2iT - T^{2}$$
71 $$1 + T^{2}$$
73 $$1 + T^{2}$$
79 $$1 + 2T + 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

−9.777662703165187287566671656296, −8.696821163695744809947302107927, −7.69533806277893102285417128286, −7.25543411897011803892344680586, −6.41633951449973542666162314434, −5.62711434928389653058091435628, −4.44484030762349950054090385695, −3.53510049815431303241984854106, −2.49888726773389951467309249421, −1.27033692199437885483573765158, 1.66337124634595529707566199574, 2.61732237927340724392963698213, 3.47715774174445045269970651000, 4.85669050903742721122450969610, 5.72494990932537743023290213341, 6.42567958369940700772505248296, 7.20254116341639988099969785528, 8.120982520058901001726358376490, 8.785094445161926091883884528579, 9.751939668581427896467730196113