# Properties

 Label 2-1323-1.1-c3-0-67 Degree $2$ Conductor $1323$ Sign $-1$ Analytic cond. $78.0595$ Root an. cond. $8.83513$ Motivic weight $3$ Arithmetic yes Rational no Primitive yes Self-dual yes Analytic rank $1$

# Related objects

## Dirichlet series

 L(s)  = 1 − 2.05·2-s − 3.75·4-s − 14.5·5-s + 24.2·8-s + 29.8·10-s − 29.8·11-s − 13.3·13-s − 19.7·16-s + 64.2·17-s − 110.·19-s + 54.5·20-s + 61.3·22-s − 19.3·23-s + 85.3·25-s + 27.4·26-s + 111.·29-s + 192.·31-s − 152.·32-s − 132.·34-s − 71.5·37-s + 227.·38-s − 351.·40-s + 277.·41-s − 178.·43-s + 112.·44-s + 39.8·46-s + 531.·47-s + ⋯
 L(s)  = 1 − 0.728·2-s − 0.469·4-s − 1.29·5-s + 1.07·8-s + 0.944·10-s − 0.817·11-s − 0.284·13-s − 0.309·16-s + 0.917·17-s − 1.33·19-s + 0.609·20-s + 0.594·22-s − 0.175·23-s + 0.682·25-s + 0.207·26-s + 0.711·29-s + 1.11·31-s − 0.844·32-s − 0.667·34-s − 0.317·37-s + 0.970·38-s − 1.38·40-s + 1.05·41-s − 0.633·43-s + 0.383·44-s + 0.127·46-s + 1.65·47-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$1323$$    =    $$3^{3} \cdot 7^{2}$$ Sign: $-1$ Analytic conductor: $$78.0595$$ Root analytic conductor: $$8.83513$$ Motivic weight: $$3$$ Rational: no Arithmetic: yes Character: Trivial Primitive: yes Self-dual: yes Analytic rank: $$1$$ Selberg data: $$(2,\ 1323,\ (\ :3/2),\ -1)$$

## Particular Values

 $$L(2)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{5}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1$$
7 $$1$$
good2 $$1 + 2.05T + 8T^{2}$$
5 $$1 + 14.5T + 125T^{2}$$
11 $$1 + 29.8T + 1.33e3T^{2}$$
13 $$1 + 13.3T + 2.19e3T^{2}$$
17 $$1 - 64.2T + 4.91e3T^{2}$$
19 $$1 + 110.T + 6.85e3T^{2}$$
23 $$1 + 19.3T + 1.21e4T^{2}$$
29 $$1 - 111.T + 2.43e4T^{2}$$
31 $$1 - 192.T + 2.97e4T^{2}$$
37 $$1 + 71.5T + 5.06e4T^{2}$$
41 $$1 - 277.T + 6.89e4T^{2}$$
43 $$1 + 178.T + 7.95e4T^{2}$$
47 $$1 - 531.T + 1.03e5T^{2}$$
53 $$1 - 310.T + 1.48e5T^{2}$$
59 $$1 + 722.T + 2.05e5T^{2}$$
61 $$1 - 663.T + 2.26e5T^{2}$$
67 $$1 + 608.T + 3.00e5T^{2}$$
71 $$1 - 976.T + 3.57e5T^{2}$$
73 $$1 - 261.T + 3.89e5T^{2}$$
79 $$1 - 1.23e3T + 4.93e5T^{2}$$
83 $$1 - 1.22e3T + 5.71e5T^{2}$$
89 $$1 + 791.T + 7.04e5T^{2}$$
97 $$1 + 935.T + 9.12e5T^{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

−8.682849696145660827140494961354, −8.004207726995912784100311856622, −7.69071843024610099870415836489, −6.63161931346524311809211519907, −5.33368599831190043425442093392, −4.46231177138341515217118411778, −3.76514500587472660682676713347, −2.48602901327919660557542862417, −0.900100721732320072677139453255, 0, 0.900100721732320072677139453255, 2.48602901327919660557542862417, 3.76514500587472660682676713347, 4.46231177138341515217118411778, 5.33368599831190043425442093392, 6.63161931346524311809211519907, 7.69071843024610099870415836489, 8.004207726995912784100311856622, 8.682849696145660827140494961354