# Properties

 Label 2-975-195.134-c0-0-1 Degree $2$ Conductor $975$ Sign $0.561 + 0.827i$ Analytic cond. $0.486588$ Root an. cond. $0.697558$ Motivic weight $0$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

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## Dirichlet series

 L(s)  = 1 + (0.866 + 0.5i)3-s + (−0.5 − 0.866i)4-s + (−0.866 − 1.5i)7-s + (0.499 + 0.866i)9-s − 0.999i·12-s − i·13-s + (−0.499 + 0.866i)16-s + (1.5 − 0.866i)19-s − 1.73i·21-s + 0.999i·27-s + (−0.866 + 1.5i)28-s + (0.499 − 0.866i)36-s + (0.5 − 0.866i)39-s + (−0.866 + 0.5i)43-s + (−0.866 + 0.499i)48-s + (−1 + 1.73i)49-s + ⋯
 L(s)  = 1 + (0.866 + 0.5i)3-s + (−0.5 − 0.866i)4-s + (−0.866 − 1.5i)7-s + (0.499 + 0.866i)9-s − 0.999i·12-s − i·13-s + (−0.499 + 0.866i)16-s + (1.5 − 0.866i)19-s − 1.73i·21-s + 0.999i·27-s + (−0.866 + 1.5i)28-s + (0.499 − 0.866i)36-s + (0.5 − 0.866i)39-s + (−0.866 + 0.5i)43-s + (−0.866 + 0.499i)48-s + (−1 + 1.73i)49-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.561 + 0.827i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.561 + 0.827i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$975$$    =    $$3 \cdot 5^{2} \cdot 13$$ Sign: $0.561 + 0.827i$ Analytic conductor: $$0.486588$$ Root analytic conductor: $$0.697558$$ Motivic weight: $$0$$ Rational: no Arithmetic: yes Character: $\chi_{975} (524, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 975,\ (\ :0),\ 0.561 + 0.827i)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$1.099363382$$ $$L(\frac12)$$ $$\approx$$ $$1.099363382$$ $$L(1)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1 + (-0.866 - 0.5i)T$$
5 $$1$$
13 $$1 + iT$$
good2 $$1 + (0.5 + 0.866i)T^{2}$$
7 $$1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2}$$
11 $$1 + (-0.5 - 0.866i)T^{2}$$
17 $$1 + (-0.5 + 0.866i)T^{2}$$
19 $$1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2}$$
23 $$1 + (-0.5 - 0.866i)T^{2}$$
29 $$1 + (0.5 + 0.866i)T^{2}$$
31 $$1 - T^{2}$$
37 $$1 + (-0.5 - 0.866i)T^{2}$$
41 $$1 + (-0.5 - 0.866i)T^{2}$$
43 $$1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2}$$
47 $$1 - T^{2}$$
53 $$1 + T^{2}$$
59 $$1 + (-0.5 + 0.866i)T^{2}$$
61 $$1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2}$$
67 $$1 + (0.866 - 1.5i)T + (-0.5 - 0.866i)T^{2}$$
71 $$1 + (-0.5 + 0.866i)T^{2}$$
73 $$1 - 1.73T + T^{2}$$
79 $$1 + T + T^{2}$$
83 $$1 - T^{2}$$
89 $$1 + (-0.5 - 0.866i)T^{2}$$
97 $$1 + (-0.5 + 0.866i)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

−10.00210678736779568252937968940, −9.511535357476655194006887620102, −8.571615789990733212624020058553, −7.54996248150242311858563962661, −6.86924625162337166888630472066, −5.59408513026664493825857412887, −4.68796205596267753559955361145, −3.77837483024939597354319271861, −2.90823150940968889979374470656, −1.05179659945309384500839421340, 2.03520041069713283106218815118, 3.08536791602852858864582493652, 3.71451700241838555798284859208, 5.09314228891923749823928011584, 6.23634886445008342863848866109, 7.08495403209077054176760137329, 8.012982025089095922207846637588, 8.684338471496467317913400998891, 9.416807763882787432333171951251, 9.746404062137856585109115297595