# Properties

 Label 2-6e4-3.2-c2-0-16 Degree $2$ Conductor $1296$ Sign $1$ Analytic cond. $35.3134$ Root an. cond. $5.94251$ Motivic weight $2$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + 1.55i·5-s − 12.3·7-s − 14.6i·11-s − 10.8·13-s + 28.9i·17-s − 3.60·19-s + 14.6i·23-s + 22.5·25-s − 28.1i·29-s − 8·31-s − 19.2i·35-s + 22.5·37-s − 25.1i·41-s + 53.1·43-s − 16.9i·47-s + ⋯
 L(s)  = 1 + 0.310i·5-s − 1.77·7-s − 1.33i·11-s − 0.831·13-s + 1.70i·17-s − 0.189·19-s + 0.638i·23-s + 0.903·25-s − 0.970i·29-s − 0.258·31-s − 0.549i·35-s + 0.609·37-s − 0.613i·41-s + 1.23·43-s − 0.361i·47-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(3-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$1296$$    =    $$2^{4} \cdot 3^{4}$$ Sign: $1$ Analytic conductor: $$35.3134$$ Root analytic conductor: $$5.94251$$ Motivic weight: $$2$$ Rational: no Arithmetic: yes Character: $\chi_{1296} (161, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 1296,\ (\ :1),\ 1)$$

## Particular Values

 $$L(\frac{3}{2})$$ $$\approx$$ $$1.160461694$$ $$L(\frac12)$$ $$\approx$$ $$1.160461694$$ $$L(2)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
3 $$1$$
good5 $$1 - 1.55iT - 25T^{2}$$
7 $$1 + 12.3T + 49T^{2}$$
11 $$1 + 14.6iT - 121T^{2}$$
13 $$1 + 10.8T + 169T^{2}$$
17 $$1 - 28.9iT - 289T^{2}$$
19 $$1 + 3.60T + 361T^{2}$$
23 $$1 - 14.6iT - 529T^{2}$$
29 $$1 + 28.1iT - 841T^{2}$$
31 $$1 + 8T + 961T^{2}$$
37 $$1 - 22.5T + 1.36e3T^{2}$$
41 $$1 + 25.1iT - 1.68e3T^{2}$$
43 $$1 - 53.1T + 1.84e3T^{2}$$
47 $$1 + 16.9iT - 2.20e3T^{2}$$
53 $$1 + 84.5iT - 2.80e3T^{2}$$
59 $$1 - 91.0iT - 3.48e3T^{2}$$
61 $$1 + 13T + 3.72e3T^{2}$$
67 $$1 - 41.1T + 4.48e3T^{2}$$
71 $$1 - 16.3iT - 5.04e3T^{2}$$
73 $$1 - 71.5T + 5.32e3T^{2}$$
79 $$1 - 46.7T + 6.24e3T^{2}$$
83 $$1 - 15.3iT - 6.88e3T^{2}$$
89 $$1 - 78.9iT - 7.92e3T^{2}$$
97 $$1 - 91.1T + 9.40e3T^{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

−9.489460901027580149829772817263, −8.761927816063163961664142065428, −7.84352384581794493081423288282, −6.82329841934416079082101531541, −6.18507873942567468096268557501, −5.55308525064078227101288899382, −3.99676020802141707732509991083, −3.34493001838919030456233567788, −2.40488931889244623305361437806, −0.59289478156634947715199915868, 0.61633107363101671411790101312, 2.40265293466052000153383278375, 3.13399330374675070948933156806, 4.43796324458463718942592496890, 5.11226257044474976653757204288, 6.33491145572029935782786890845, 7.01452567492491256209256858840, 7.57873001370153118050136110382, 9.032267973163003126985144554510, 9.452213195057428505935304298205