# Properties

 Label 2-3267-297.113-c0-0-0 Degree $2$ Conductor $3267$ Sign $-0.988 - 0.150i$ Analytic cond. $1.63044$ Root an. cond. $1.27688$ Motivic weight $0$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

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## Dirichlet series

 L(s)  = 1 + (0.374 − 0.927i)3-s + (−0.882 − 0.469i)4-s + (−0.542 + 1.89i)5-s + (−0.719 − 0.694i)9-s + (−0.766 + 0.642i)12-s + (1.55 + 1.21i)15-s + (0.559 + 0.829i)16-s + (1.36 − 1.41i)20-s + (−1.70 − 0.300i)23-s + (−2.44 − 1.52i)25-s + (−0.913 + 0.406i)27-s + (−0.671 − 1.37i)31-s + (0.309 + 0.951i)36-s + (−0.232 − 0.258i)37-s + (1.70 − 0.984i)45-s + ⋯
 L(s)  = 1 + (0.374 − 0.927i)3-s + (−0.882 − 0.469i)4-s + (−0.542 + 1.89i)5-s + (−0.719 − 0.694i)9-s + (−0.766 + 0.642i)12-s + (1.55 + 1.21i)15-s + (0.559 + 0.829i)16-s + (1.36 − 1.41i)20-s + (−1.70 − 0.300i)23-s + (−2.44 − 1.52i)25-s + (−0.913 + 0.406i)27-s + (−0.671 − 1.37i)31-s + (0.309 + 0.951i)36-s + (−0.232 − 0.258i)37-s + (1.70 − 0.984i)45-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 3267 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.988 - 0.150i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 3267 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.988 - 0.150i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$3267$$    =    $$3^{3} \cdot 11^{2}$$ Sign: $-0.988 - 0.150i$ Analytic conductor: $$1.63044$$ Root analytic conductor: $$1.27688$$ Motivic weight: $$0$$ Rational: no Arithmetic: yes Character: $\chi_{3267} (2786, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 3267,\ (\ :0),\ -0.988 - 0.150i)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$0.04519643734$$ $$L(\frac12)$$ $$\approx$$ $$0.04519643734$$ $$L(1)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1 + (-0.374 + 0.927i)T$$
11 $$1$$
good2 $$1 + (0.882 + 0.469i)T^{2}$$
5 $$1 + (0.542 - 1.89i)T + (-0.848 - 0.529i)T^{2}$$
7 $$1 + (0.961 + 0.275i)T^{2}$$
13 $$1 + (0.990 + 0.139i)T^{2}$$
17 $$1 + (-0.669 + 0.743i)T^{2}$$
19 $$1 + (-0.104 - 0.994i)T^{2}$$
23 $$1 + (1.70 + 0.300i)T + (0.939 + 0.342i)T^{2}$$
29 $$1 + (0.719 - 0.694i)T^{2}$$
31 $$1 + (0.671 + 1.37i)T + (-0.615 + 0.788i)T^{2}$$
37 $$1 + (0.232 + 0.258i)T + (-0.104 + 0.994i)T^{2}$$
41 $$1 + (0.719 + 0.694i)T^{2}$$
43 $$1 + (0.766 + 0.642i)T^{2}$$
47 $$1 + (-0.321 - 0.603i)T + (-0.559 + 0.829i)T^{2}$$
53 $$1 + (-0.755 - 1.04i)T + (-0.309 + 0.951i)T^{2}$$
59 $$1 + (1.28 - 0.0448i)T + (0.997 - 0.0697i)T^{2}$$
61 $$1 + (-0.615 - 0.788i)T^{2}$$
67 $$1 + (1.17 + 0.984i)T + (0.173 + 0.984i)T^{2}$$
71 $$1 + (0.278 + 0.624i)T + (-0.669 + 0.743i)T^{2}$$
73 $$1 + (0.913 - 0.406i)T^{2}$$
79 $$1 + (-0.882 - 0.469i)T^{2}$$
83 $$1 + (-0.990 + 0.139i)T^{2}$$
89 $$1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2}$$
97 $$1 + (1.80 - 0.518i)T + (0.848 - 0.529i)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

−8.120647554799933037995466130544, −7.77264892136879581961604283146, −7.00739933978817343240116885277, −6.15125789695300628384668773094, −5.83574502119993258235836196288, −4.27819190191630532555512585522, −3.64203059581268271444200054209, −2.74253671099325749754971100913, −1.84010405668305246494044459179, −0.02576460796519395051853795457, 1.62508953193746238429495315226, 3.22556132019212632215869645704, 4.03383807456461692156366025750, 4.43700038984622771974940066194, 5.21459587897563653451881810478, 5.70525730314851929544015929737, 7.37431797546830768770370943500, 8.213695814766061714417090871970, 8.450526665275352277249714900577, 9.103343717788894458027414638039