# Properties

 Label 2-12e3-1.1-c3-0-2 Degree $2$ Conductor $1728$ Sign $1$ Analytic cond. $101.955$ Root an. cond. $10.0972$ Motivic weight $3$ Arithmetic yes Rational no Primitive yes Self-dual yes Analytic rank $0$

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## Dirichlet series

 L(s)  = 1 − 15.4·5-s − 23.8·7-s − 14.2·11-s + 13.8·13-s + 80.5·17-s − 144.·19-s − 141.·23-s + 112.·25-s − 251.·29-s + 16.6·31-s + 367.·35-s − 305.·37-s − 429.·41-s − 181.·43-s − 79.4·47-s + 225.·49-s − 663.·53-s + 219.·55-s − 220.·59-s + 473.·61-s − 213.·65-s − 647.·67-s + 14.4·71-s + 776.·73-s + 339.·77-s + 257.·79-s − 1.28e3·83-s + ⋯
 L(s)  = 1 − 1.37·5-s − 1.28·7-s − 0.390·11-s + 0.295·13-s + 1.14·17-s − 1.74·19-s − 1.27·23-s + 0.901·25-s − 1.60·29-s + 0.0965·31-s + 1.77·35-s − 1.35·37-s − 1.63·41-s − 0.644·43-s − 0.246·47-s + 0.655·49-s − 1.72·53-s + 0.538·55-s − 0.486·59-s + 0.993·61-s − 0.406·65-s − 1.18·67-s + 0.0242·71-s + 1.24·73-s + 0.502·77-s + 0.367·79-s − 1.69·83-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$1728$$    =    $$2^{6} \cdot 3^{3}$$ Sign: $1$ Analytic conductor: $$101.955$$ Root analytic conductor: $$10.0972$$ Motivic weight: $$3$$ Rational: no Arithmetic: yes Character: $\chi_{1728} (1, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(2,\ 1728,\ (\ :3/2),\ 1)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$0.1186181059$$ $$L(\frac12)$$ $$\approx$$ $$0.1186181059$$ $$L(\frac{5}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
3 $$1$$
good5 $$1 + 15.4T + 125T^{2}$$
7 $$1 + 23.8T + 343T^{2}$$
11 $$1 + 14.2T + 1.33e3T^{2}$$
13 $$1 - 13.8T + 2.19e3T^{2}$$
17 $$1 - 80.5T + 4.91e3T^{2}$$
19 $$1 + 144.T + 6.85e3T^{2}$$
23 $$1 + 141.T + 1.21e4T^{2}$$
29 $$1 + 251.T + 2.43e4T^{2}$$
31 $$1 - 16.6T + 2.97e4T^{2}$$
37 $$1 + 305.T + 5.06e4T^{2}$$
41 $$1 + 429.T + 6.89e4T^{2}$$
43 $$1 + 181.T + 7.95e4T^{2}$$
47 $$1 + 79.4T + 1.03e5T^{2}$$
53 $$1 + 663.T + 1.48e5T^{2}$$
59 $$1 + 220.T + 2.05e5T^{2}$$
61 $$1 - 473.T + 2.26e5T^{2}$$
67 $$1 + 647.T + 3.00e5T^{2}$$
71 $$1 - 14.4T + 3.57e5T^{2}$$
73 $$1 - 776.T + 3.89e5T^{2}$$
79 $$1 - 257.T + 4.93e5T^{2}$$
83 $$1 + 1.28e3T + 5.71e5T^{2}$$
89 $$1 - 156.T + 7.04e5T^{2}$$
97 $$1 - 1.16e3T + 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

−8.797001251532814768053227947356, −8.146562049973368067210063741889, −7.45288748471507346510946204860, −6.59264484946358436920412706658, −5.86420357352723047894603606616, −4.71110847689193430532551465012, −3.59322894492606852207852726764, −3.42665206386592590894863730564, −1.91065763311010522864532296909, −0.15447576520705160532441018235, 0.15447576520705160532441018235, 1.91065763311010522864532296909, 3.42665206386592590894863730564, 3.59322894492606852207852726764, 4.71110847689193430532551465012, 5.86420357352723047894603606616, 6.59264484946358436920412706658, 7.45288748471507346510946204860, 8.146562049973368067210063741889, 8.797001251532814768053227947356