# Properties

 Degree $2$ Conductor $1575$ Sign $0.392 + 0.919i$ Motivic weight $0$ Primitive yes Self-dual no Analytic rank $0$

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## Dirichlet series

 L(s)  = 1 + (−0.258 − 0.965i)2-s + (0.707 + 0.707i)3-s + (0.500 − 0.866i)6-s + (0.707 − 0.707i)7-s + (−0.707 − 0.707i)8-s + 1.00i·9-s + (0.965 − 0.258i)13-s + (−0.866 − 0.500i)14-s + (−0.5 + 0.866i)16-s + (−0.258 − 0.965i)17-s + (0.965 − 0.258i)18-s + (−0.866 + 0.5i)19-s + 1.00·21-s − 0.999i·24-s + (−0.499 − 0.866i)26-s + (−0.707 + 0.707i)27-s + ⋯
 L(s)  = 1 + (−0.258 − 0.965i)2-s + (0.707 + 0.707i)3-s + (0.500 − 0.866i)6-s + (0.707 − 0.707i)7-s + (−0.707 − 0.707i)8-s + 1.00i·9-s + (0.965 − 0.258i)13-s + (−0.866 − 0.500i)14-s + (−0.5 + 0.866i)16-s + (−0.258 − 0.965i)17-s + (0.965 − 0.258i)18-s + (−0.866 + 0.5i)19-s + 1.00·21-s − 0.999i·24-s + (−0.499 − 0.866i)26-s + (−0.707 + 0.707i)27-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.392 + 0.919i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.392 + 0.919i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$1575$$    =    $$3^{2} \cdot 5^{2} \cdot 7$$ Sign: $0.392 + 0.919i$ Motivic weight: $$0$$ Character: $\chi_{1575} (193, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 1575,\ (\ :0),\ 0.392 + 0.919i)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$1.410098365$$ $$L(\frac12)$$ $$\approx$$ $$1.410098365$$ $$L(1)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1 + (-0.707 - 0.707i)T$$
5 $$1$$
7 $$1 + (-0.707 + 0.707i)T$$
good2 $$1 + (0.258 + 0.965i)T + (-0.866 + 0.5i)T^{2}$$
11 $$1 + T^{2}$$
13 $$1 + (-0.965 + 0.258i)T + (0.866 - 0.5i)T^{2}$$
17 $$1 + (0.258 + 0.965i)T + (-0.866 + 0.5i)T^{2}$$
19 $$1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2}$$
23 $$1 + iT^{2}$$
29 $$1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2}$$
31 $$1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2}$$
37 $$1 + (0.965 + 0.258i)T + (0.866 + 0.5i)T^{2}$$
41 $$1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2}$$
43 $$1 + (-0.258 + 0.965i)T + (-0.866 - 0.5i)T^{2}$$
47 $$1 + (-0.258 - 0.965i)T + (-0.866 + 0.5i)T^{2}$$
53 $$1 + (0.965 - 0.258i)T + (0.866 - 0.5i)T^{2}$$
59 $$1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2}$$
61 $$1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2}$$
67 $$1 + (-0.965 - 0.258i)T + (0.866 + 0.5i)T^{2}$$
71 $$1 + 2T + T^{2}$$
73 $$1 + (-0.965 + 0.258i)T + (0.866 - 0.5i)T^{2}$$
79 $$1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2}$$
83 $$1 + (0.258 - 0.965i)T + (-0.866 - 0.5i)T^{2}$$
89 $$1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2}$$
97 $$1 + (0.965 + 0.258i)T + (0.866 + 0.5i)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

−9.674533377327000463156271679046, −8.756896816610440773430961172423, −8.250527551430187550985831276307, −7.21595901499485996283343328513, −6.25812986912009067542057710429, −5.02183334435202348134933835355, −4.14272414368454757616696711833, −3.35988578309785708866730364071, −2.40365667404626328749019836380, −1.31296636736101174825138448459, 1.71710089531194582892474345443, 2.59423705567609324146549979766, 3.77266632992031315665667382987, 5.03956849871514225659937931066, 6.21907595265765632459265590765, 6.45369490331203329685257398117, 7.47487465714664736637391474876, 8.296591136709525968916893599602, 8.583226305235080019589441328021, 9.189164220187618661665505634636