# Properties

 Label 2-165-165.98-c2-0-9 Degree $2$ Conductor $165$ Sign $0.763 - 0.646i$ Analytic cond. $4.49592$ Root an. cond. $2.12035$ Motivic weight $2$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (1.58 − 1.58i)2-s − 3·3-s − 1.00i·4-s + (−4 + 3i)5-s + (−4.74 + 4.74i)6-s + (3.16 + 3.16i)7-s + (4.74 + 4.74i)8-s + 9·9-s + (−1.58 + 11.0i)10-s + (6.32 + 9i)11-s + 3.00i·12-s + (3.16 − 3.16i)13-s + 10.0·14-s + (12 − 9i)15-s + 19·16-s + (−22.1 + 22.1i)17-s + ⋯
 L(s)  = 1 + (0.790 − 0.790i)2-s − 3-s − 0.250i·4-s + (−0.800 + 0.600i)5-s + (−0.790 + 0.790i)6-s + (0.451 + 0.451i)7-s + (0.592 + 0.592i)8-s + 9-s + (−0.158 + 1.10i)10-s + (0.574 + 0.818i)11-s + 0.250i·12-s + (0.243 − 0.243i)13-s + 0.714·14-s + (0.800 − 0.599i)15-s + 1.18·16-s + (−1.30 + 1.30i)17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.763 - 0.646i)\, \overline{\Lambda}(3-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.763 - 0.646i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$165$$    =    $$3 \cdot 5 \cdot 11$$ Sign: $0.763 - 0.646i$ Analytic conductor: $$4.49592$$ Root analytic conductor: $$2.12035$$ Motivic weight: $$2$$ Rational: no Arithmetic: yes Character: $\chi_{165} (98, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 165,\ (\ :1),\ 0.763 - 0.646i)$$

## Particular Values

 $$L(\frac{3}{2})$$ $$\approx$$ $$1.31678 + 0.482515i$$ $$L(\frac12)$$ $$\approx$$ $$1.31678 + 0.482515i$$ $$L(2)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1 + 3T$$
5 $$1 + (4 - 3i)T$$
11 $$1 + (-6.32 - 9i)T$$
good2 $$1 + (-1.58 + 1.58i)T - 4iT^{2}$$
7 $$1 + (-3.16 - 3.16i)T + 49iT^{2}$$
13 $$1 + (-3.16 + 3.16i)T - 169iT^{2}$$
17 $$1 + (22.1 - 22.1i)T - 289iT^{2}$$
19 $$1 + 12.6T + 361T^{2}$$
23 $$1 + (7 - 7i)T - 529iT^{2}$$
29 $$1 - 18.9iT - 841T^{2}$$
31 $$1 - 20T + 961T^{2}$$
37 $$1 + (-7 + 7i)T - 1.36e3iT^{2}$$
41 $$1 - 69.5T + 1.68e3T^{2}$$
43 $$1 + (-22.1 + 22.1i)T - 1.84e3iT^{2}$$
47 $$1 + (43 + 43i)T + 2.20e3iT^{2}$$
53 $$1 + (-17 + 17i)T - 2.80e3iT^{2}$$
59 $$1 - 22T + 3.48e3T^{2}$$
61 $$1 - 94.8iT - 3.72e3T^{2}$$
67 $$1 + (47 - 47i)T - 4.48e3iT^{2}$$
71 $$1 + 120iT - 5.04e3T^{2}$$
73 $$1 + (-22.1 + 22.1i)T - 5.32e3iT^{2}$$
79 $$1 - 6.32T + 6.24e3T^{2}$$
83 $$1 + (60.0 + 60.0i)T + 6.88e3iT^{2}$$
89 $$1 - 100T + 7.92e3T^{2}$$
97 $$1 + (-43 + 43i)T - 9.40e3iT^{2}$$
show less
$$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

−12.44213327061526295494684933109, −11.77059973151490513271338326332, −11.02578689445890207508237521204, −10.36094982041398778292358712897, −8.534092574162266787610122856903, −7.28698936576666739582535603796, −6.10669928088490121067431041171, −4.62024092154609501269514156275, −3.90314218138986698128611897924, −2.02618575779198540181228204384, 0.815873772553838929220468044392, 4.20711322060649126654161278680, 4.66987124642198007116386307474, 6.00951771345949351299846562162, 6.88018361450821282193794470153, 7.969917895895861624085367761883, 9.401910680671323001493922887779, 10.94458171316355277358844700442, 11.45186800212794118935670964269, 12.61970814079493498851470502086