# Properties

 Label 2-39e2-1.1-c3-0-148 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 $1$

# Related objects

## Dirichlet series

 L(s)  = 1 − 3.66·2-s + 5.43·4-s + 9.45·5-s + 15.4·7-s + 9.39·8-s − 34.6·10-s − 12.7·11-s − 56.4·14-s − 77.9·16-s − 18.9·17-s + 95.4·19-s + 51.4·20-s + 46.6·22-s + 104.·23-s − 35.6·25-s + 83.8·28-s + 23.3·29-s − 177.·31-s + 210.·32-s + 69.6·34-s + 145.·35-s − 350.·37-s − 350.·38-s + 88.7·40-s − 348.·41-s + 60.8·43-s − 69.2·44-s + ⋯
 L(s)  = 1 − 1.29·2-s + 0.679·4-s + 0.845·5-s + 0.832·7-s + 0.415·8-s − 1.09·10-s − 0.348·11-s − 1.07·14-s − 1.21·16-s − 0.271·17-s + 1.15·19-s + 0.574·20-s + 0.452·22-s + 0.945·23-s − 0.284·25-s + 0.565·28-s + 0.149·29-s − 1.02·31-s + 1.16·32-s + 0.351·34-s + 0.703·35-s − 1.55·37-s − 1.49·38-s + 0.351·40-s − 1.32·41-s + 0.215·43-s − 0.237·44-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: $$1$$ Selberg data: $$(2,\ 1521,\ (\ :3/2),\ -1)$$

## 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$$
13 $$1$$
good2 $$1 + 3.66T + 8T^{2}$$
5 $$1 - 9.45T + 125T^{2}$$
7 $$1 - 15.4T + 343T^{2}$$
11 $$1 + 12.7T + 1.33e3T^{2}$$
17 $$1 + 18.9T + 4.91e3T^{2}$$
19 $$1 - 95.4T + 6.85e3T^{2}$$
23 $$1 - 104.T + 1.21e4T^{2}$$
29 $$1 - 23.3T + 2.43e4T^{2}$$
31 $$1 + 177.T + 2.97e4T^{2}$$
37 $$1 + 350.T + 5.06e4T^{2}$$
41 $$1 + 348.T + 6.89e4T^{2}$$
43 $$1 - 60.8T + 7.95e4T^{2}$$
47 $$1 + 226.T + 1.03e5T^{2}$$
53 $$1 + 294.T + 1.48e5T^{2}$$
59 $$1 + 596.T + 2.05e5T^{2}$$
61 $$1 + 487.T + 2.26e5T^{2}$$
67 $$1 + 943.T + 3.00e5T^{2}$$
71 $$1 - 1.04e3T + 3.57e5T^{2}$$
73 $$1 + 554.T + 3.89e5T^{2}$$
79 $$1 - 126.T + 4.93e5T^{2}$$
83 $$1 - 1.44e3T + 5.71e5T^{2}$$
89 $$1 + 247.T + 7.04e5T^{2}$$
97 $$1 - 136.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

−8.896552912105502014684254008790, −7.975370953055441535160446786446, −7.40434605241084705183376393378, −6.50064403802197746081178154187, −5.32057554153259172178884349756, −4.77646928878874776091293517381, −3.25748067082039095176295830528, −1.93081365299388562573594371294, −1.36391205226987486293852414099, 0, 1.36391205226987486293852414099, 1.93081365299388562573594371294, 3.25748067082039095176295830528, 4.77646928878874776091293517381, 5.32057554153259172178884349756, 6.50064403802197746081178154187, 7.40434605241084705183376393378, 7.975370953055441535160446786446, 8.896552912105502014684254008790