# Properties

 Label 2-39e2-1.1-c3-0-74 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 $0$

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## Dirichlet series

 L(s)  = 1 + 4.83·2-s + 15.4·4-s − 21.1·5-s + 16.2·7-s + 35.8·8-s − 102.·10-s − 30.7·11-s + 78.7·14-s + 49.9·16-s − 46.2·17-s + 144.·19-s − 326.·20-s − 148.·22-s − 8.38·23-s + 324.·25-s + 250.·28-s + 242.·29-s + 87.9·31-s − 44.5·32-s − 223.·34-s − 345.·35-s − 49.6·37-s + 700.·38-s − 758.·40-s + 107.·41-s − 35.4·43-s − 473.·44-s + ⋯
 L(s)  = 1 + 1.71·2-s + 1.92·4-s − 1.89·5-s + 0.879·7-s + 1.58·8-s − 3.24·10-s − 0.842·11-s + 1.50·14-s + 0.781·16-s − 0.659·17-s + 1.74·19-s − 3.65·20-s − 1.44·22-s − 0.0759·23-s + 2.59·25-s + 1.69·28-s + 1.55·29-s + 0.509·31-s − 0.246·32-s − 1.12·34-s − 1.66·35-s − 0.220·37-s + 2.99·38-s − 3.00·40-s + 0.411·41-s − 0.125·43-s − 1.62·44-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

 $$L(2)$$ $$\approx$$ $$4.689089025$$ $$L(\frac12)$$ $$\approx$$ $$4.689089025$$ $$L(\frac{5}{2})$$ 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 - 4.83T + 8T^{2}$$
5 $$1 + 21.1T + 125T^{2}$$
7 $$1 - 16.2T + 343T^{2}$$
11 $$1 + 30.7T + 1.33e3T^{2}$$
17 $$1 + 46.2T + 4.91e3T^{2}$$
19 $$1 - 144.T + 6.85e3T^{2}$$
23 $$1 + 8.38T + 1.21e4T^{2}$$
29 $$1 - 242.T + 2.43e4T^{2}$$
31 $$1 - 87.9T + 2.97e4T^{2}$$
37 $$1 + 49.6T + 5.06e4T^{2}$$
41 $$1 - 107.T + 6.89e4T^{2}$$
43 $$1 + 35.4T + 7.95e4T^{2}$$
47 $$1 - 374.T + 1.03e5T^{2}$$
53 $$1 - 348.T + 1.48e5T^{2}$$
59 $$1 - 679.T + 2.05e5T^{2}$$
61 $$1 + 230.T + 2.26e5T^{2}$$
67 $$1 - 295.T + 3.00e5T^{2}$$
71 $$1 - 329.T + 3.57e5T^{2}$$
73 $$1 + 48.9T + 3.89e5T^{2}$$
79 $$1 + 107.T + 4.93e5T^{2}$$
83 $$1 - 515.T + 5.71e5T^{2}$$
89 $$1 - 984.T + 7.04e5T^{2}$$
97 $$1 + 487.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

−8.772034470157670565834545711152, −7.947173352026690852102283196783, −7.41872432632305289112572900908, −6.65113316257860099837312111453, −5.35947068080819761808902964107, −4.81282938106690967396580788092, −4.15389331026428573316859791658, −3.31691998835778381191216915854, −2.51498228928534835932888620815, −0.815268484858022827544365020368, 0.815268484858022827544365020368, 2.51498228928534835932888620815, 3.31691998835778381191216915854, 4.15389331026428573316859791658, 4.81282938106690967396580788092, 5.35947068080819761808902964107, 6.65113316257860099837312111453, 7.41872432632305289112572900908, 7.947173352026690852102283196783, 8.772034470157670565834545711152