# Properties

 Label 2-1911-273.233-c0-0-11 Degree $2$ Conductor $1911$ Sign $-0.749 + 0.661i$ Analytic cond. $0.953713$ Root an. cond. $0.976582$ Motivic weight $0$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

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## Dirichlet series

 L(s)  = 1 + (0.382 − 0.662i)2-s + (−0.5 − 0.866i)3-s + (0.207 + 0.358i)4-s + (0.923 − 1.60i)5-s − 0.765·6-s + 1.08·8-s + (−0.499 + 0.866i)9-s + (−0.707 − 1.22i)10-s + (−0.382 − 0.662i)11-s + (0.207 − 0.358i)12-s − 13-s − 1.84·15-s + (0.207 − 0.358i)16-s + (0.382 + 0.662i)18-s + 0.765·20-s + ⋯
 L(s)  = 1 + (0.382 − 0.662i)2-s + (−0.5 − 0.866i)3-s + (0.207 + 0.358i)4-s + (0.923 − 1.60i)5-s − 0.765·6-s + 1.08·8-s + (−0.499 + 0.866i)9-s + (−0.707 − 1.22i)10-s + (−0.382 − 0.662i)11-s + (0.207 − 0.358i)12-s − 13-s − 1.84·15-s + (0.207 − 0.358i)16-s + (0.382 + 0.662i)18-s + 0.765·20-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.749 + 0.661i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.749 + 0.661i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$1911$$    =    $$3 \cdot 7^{2} \cdot 13$$ Sign: $-0.749 + 0.661i$ Analytic conductor: $$0.953713$$ Root analytic conductor: $$0.976582$$ Motivic weight: $$0$$ Rational: no Arithmetic: yes Character: $\chi_{1911} (1598, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 1911,\ (\ :0),\ -0.749 + 0.661i)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$1.449917808$$ $$L(\frac12)$$ $$\approx$$ $$1.449917808$$ $$L(1)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1 + (0.5 + 0.866i)T$$
7 $$1$$
13 $$1 + T$$
good2 $$1 + (-0.382 + 0.662i)T + (-0.5 - 0.866i)T^{2}$$
5 $$1 + (-0.923 + 1.60i)T + (-0.5 - 0.866i)T^{2}$$
11 $$1 + (0.382 + 0.662i)T + (-0.5 + 0.866i)T^{2}$$
17 $$1 + (0.5 - 0.866i)T^{2}$$
19 $$1 + (0.5 + 0.866i)T^{2}$$
23 $$1 + (0.5 + 0.866i)T^{2}$$
29 $$1 - T^{2}$$
31 $$1 + (0.5 - 0.866i)T^{2}$$
37 $$1 + (0.5 + 0.866i)T^{2}$$
41 $$1 - 1.84T + T^{2}$$
43 $$1 + T^{2}$$
47 $$1 + (0.923 - 1.60i)T + (-0.5 - 0.866i)T^{2}$$
53 $$1 + (0.5 - 0.866i)T^{2}$$
59 $$1 + (0.382 + 0.662i)T + (-0.5 + 0.866i)T^{2}$$
61 $$1 + (-0.5 - 0.866i)T^{2}$$
67 $$1 + (0.5 - 0.866i)T^{2}$$
71 $$1 - 1.84T + T^{2}$$
73 $$1 + (0.5 - 0.866i)T^{2}$$
79 $$1 + (0.707 - 1.22i)T + (-0.5 - 0.866i)T^{2}$$
83 $$1 + 0.765T + T^{2}$$
89 $$1 + (-0.382 + 0.662i)T + (-0.5 - 0.866i)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

−9.123578083768688105179514188344, −8.142932349533341060397994479852, −7.73643768018726526098922996859, −6.60707245181531177701951266639, −5.70844900249492314077503060587, −5.04949052705041863000786148130, −4.33889744536100124009003410657, −2.80085278337448965096891717605, −2.01993222046230178137897736561, −1.02268019764110511270924881952, 2.05611580767749605967054170809, 2.92375765469683277407709145621, 4.16973415007004942936615968804, 5.13256260481903109531878663629, 5.69622956218373760390498443685, 6.48696293143417147297671861677, 7.01092822703179509628583612825, 7.75200933715683022700033641635, 9.294800571696405405896255204477, 9.950756835600683855745170909590