# Properties

 Label 2-39e2-1.1-c3-0-30 Degree $2$ Conductor $1521$ Sign $1$ Analytic cond. $89.7419$ Root an. cond. $9.47322$ Motivic weight $3$ Arithmetic yes Rational no Primitive yes Self-dual yes Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + 3.17·2-s + 2.07·4-s − 6.74·5-s − 14.1·7-s − 18.8·8-s − 21.3·10-s − 62.4·11-s − 44.9·14-s − 76.2·16-s + 58.6·17-s + 64.1·19-s − 13.9·20-s − 198.·22-s − 10.9·23-s − 79.5·25-s − 29.3·28-s − 216.·29-s − 38.6·31-s − 91.5·32-s + 186.·34-s + 95.5·35-s + 423.·37-s + 203.·38-s + 126.·40-s + 366.·41-s − 128.·43-s − 129.·44-s + ⋯
 L(s)  = 1 + 1.12·2-s + 0.258·4-s − 0.602·5-s − 0.765·7-s − 0.831·8-s − 0.676·10-s − 1.71·11-s − 0.858·14-s − 1.19·16-s + 0.836·17-s + 0.774·19-s − 0.156·20-s − 1.92·22-s − 0.0990·23-s − 0.636·25-s − 0.198·28-s − 1.38·29-s − 0.223·31-s − 0.505·32-s + 0.938·34-s + 0.461·35-s + 1.88·37-s + 0.869·38-s + 0.501·40-s + 1.39·41-s − 0.455·43-s − 0.443·44-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$1521$$    =    $$3^{2} \cdot 13^{2}$$ Sign: $1$ Analytic conductor: $$89.7419$$ Root analytic conductor: $$9.47322$$ Motivic weight: $$3$$ Rational: no Arithmetic: yes Character: Trivial Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(2,\ 1521,\ (\ :3/2),\ 1)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$1.496190744$$ $$L(\frac12)$$ $$\approx$$ $$1.496190744$$ $$L(\frac{5}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1$$
13 $$1$$
good2 $$1 - 3.17T + 8T^{2}$$
5 $$1 + 6.74T + 125T^{2}$$
7 $$1 + 14.1T + 343T^{2}$$
11 $$1 + 62.4T + 1.33e3T^{2}$$
17 $$1 - 58.6T + 4.91e3T^{2}$$
19 $$1 - 64.1T + 6.85e3T^{2}$$
23 $$1 + 10.9T + 1.21e4T^{2}$$
29 $$1 + 216.T + 2.43e4T^{2}$$
31 $$1 + 38.6T + 2.97e4T^{2}$$
37 $$1 - 423.T + 5.06e4T^{2}$$
41 $$1 - 366.T + 6.89e4T^{2}$$
43 $$1 + 128.T + 7.95e4T^{2}$$
47 $$1 - 93.1T + 1.03e5T^{2}$$
53 $$1 + 131.T + 1.48e5T^{2}$$
59 $$1 + 386.T + 2.05e5T^{2}$$
61 $$1 + 621.T + 2.26e5T^{2}$$
67 $$1 - 865.T + 3.00e5T^{2}$$
71 $$1 - 607.T + 3.57e5T^{2}$$
73 $$1 - 980.T + 3.89e5T^{2}$$
79 $$1 - 1.33e3T + 4.93e5T^{2}$$
83 $$1 + 907.T + 5.71e5T^{2}$$
89 $$1 - 1.03e3T + 7.04e5T^{2}$$
97 $$1 - 1.04e3T + 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

−9.333060736249340697053166230592, −7.86803806342114058061372514246, −7.70100743409639508573327481737, −6.37015536736147715622613786402, −5.61255997792977407906769926749, −5.00300040230651945125646844415, −3.93683702909906053686962607618, −3.26254144575299950498854489098, −2.44524840430735815530352293612, −0.47455874685598872431166464370, 0.47455874685598872431166464370, 2.44524840430735815530352293612, 3.26254144575299950498854489098, 3.93683702909906053686962607618, 5.00300040230651945125646844415, 5.61255997792977407906769926749, 6.37015536736147715622613786402, 7.70100743409639508573327481737, 7.86803806342114058061372514246, 9.333060736249340697053166230592