# Properties

 Label 2-33-33.32-c5-0-17 Degree $2$ Conductor $33$ Sign $-0.484 + 0.874i$ Analytic cond. $5.29266$ Root an. cond. $2.30057$ Motivic weight $5$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + 3.58·2-s + (0.133 − 15.5i)3-s − 19.1·4-s − 11.8i·5-s + (0.480 − 55.9i)6-s − 150. i·7-s − 183.·8-s + (−242. − 4.17i)9-s − 42.6i·10-s + (352. + 191. i)11-s + (−2.56 + 297. i)12-s − 778. i·13-s − 541. i·14-s + (−185. − 1.59i)15-s − 47.1·16-s + 1.25e3·17-s + ⋯
 L(s)  = 1 + 0.634·2-s + (0.00859 − 0.999i)3-s − 0.597·4-s − 0.212i·5-s + (0.00545 − 0.634i)6-s − 1.16i·7-s − 1.01·8-s + (−0.999 − 0.0171i)9-s − 0.134i·10-s + (0.878 + 0.477i)11-s + (−0.00513 + 0.597i)12-s − 1.27i·13-s − 0.738i·14-s + (−0.212 − 0.00182i)15-s − 0.0460·16-s + 1.05·17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.484 + 0.874i)\, \overline{\Lambda}(6-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.484 + 0.874i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$33$$    =    $$3 \cdot 11$$ Sign: $-0.484 + 0.874i$ Analytic conductor: $$5.29266$$ Root analytic conductor: $$2.30057$$ Motivic weight: $$5$$ Rational: no Arithmetic: yes Character: $\chi_{33} (32, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 33,\ (\ :5/2),\ -0.484 + 0.874i)$$

## Particular Values

 $$L(3)$$ $$\approx$$ $$0.786381 - 1.33523i$$ $$L(\frac12)$$ $$\approx$$ $$0.786381 - 1.33523i$$ $$L(\frac{7}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1 + (-0.133 + 15.5i)T$$
11 $$1 + (-352. - 191. i)T$$
good2 $$1 - 3.58T + 32T^{2}$$
5 $$1 + 11.8iT - 3.12e3T^{2}$$
7 $$1 + 150. iT - 1.68e4T^{2}$$
13 $$1 + 778. iT - 3.71e5T^{2}$$
17 $$1 - 1.25e3T + 1.41e6T^{2}$$
19 $$1 - 1.21e3iT - 2.47e6T^{2}$$
23 $$1 + 880. iT - 6.43e6T^{2}$$
29 $$1 - 3.20e3T + 2.05e7T^{2}$$
31 $$1 + 6.44e3T + 2.86e7T^{2}$$
37 $$1 + 9.31e3T + 6.93e7T^{2}$$
41 $$1 - 5.39e3T + 1.15e8T^{2}$$
43 $$1 + 2.08e4iT - 1.47e8T^{2}$$
47 $$1 + 1.12e4iT - 2.29e8T^{2}$$
53 $$1 + 2.39e4iT - 4.18e8T^{2}$$
59 $$1 - 2.29e4iT - 7.14e8T^{2}$$
61 $$1 - 2.72e4iT - 8.44e8T^{2}$$
67 $$1 - 2.02e4T + 1.35e9T^{2}$$
71 $$1 - 5.64e4iT - 1.80e9T^{2}$$
73 $$1 - 1.20e4iT - 2.07e9T^{2}$$
79 $$1 - 1.10e4iT - 3.07e9T^{2}$$
83 $$1 + 5.15e4T + 3.93e9T^{2}$$
89 $$1 - 8.41e4iT - 5.58e9T^{2}$$
97 $$1 - 9.93e4T + 8.58e9T^{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

−14.70835301571275326264406910870, −13.99609418752264290236294403721, −12.87709192204423397831630777483, −12.15086329126730888674533748452, −10.27039048652998048748959630616, −8.531632501272147968920210245166, −7.13221231607963175690306113095, −5.47693798597470759334405542154, −3.62450022703965031050779871180, −0.841618810941280350611269483588, 3.25960849837752797013653804291, 4.77002155527179426894864912161, 6.07889565597513368577601080630, 8.812643472837650067409725686787, 9.434420591874846042548239166354, 11.31538767511600109603705700156, 12.33954678408143323784139513844, 14.10862233589288455629614118862, 14.64615522257171952844736039514, 15.86344447432515370667338385924