# Properties

 Label 2-1452-1.1-c3-0-44 Degree $2$ Conductor $1452$ Sign $-1$ Analytic cond. $85.6707$ Root an. cond. $9.25585$ Motivic weight $3$ Arithmetic yes Rational no Primitive yes Self-dual yes Analytic rank $1$

# Related objects

## Dirichlet series

 L(s)  = 1 + 3·3-s − 3.88·5-s − 1.97·7-s + 9·9-s − 48.0·13-s − 11.6·15-s − 8.31·17-s + 102.·19-s − 5.92·21-s + 108.·23-s − 109.·25-s + 27·27-s − 88.1·29-s + 71.7·31-s + 7.67·35-s + 203.·37-s − 144.·39-s − 177.·41-s − 103.·43-s − 34.9·45-s + 185.·47-s − 339.·49-s − 24.9·51-s − 698.·53-s + 306.·57-s − 157.·59-s − 207.·61-s + ⋯
 L(s)  = 1 + 0.577·3-s − 0.347·5-s − 0.106·7-s + 0.333·9-s − 1.02·13-s − 0.200·15-s − 0.118·17-s + 1.23·19-s − 0.0616·21-s + 0.979·23-s − 0.879·25-s + 0.192·27-s − 0.564·29-s + 0.415·31-s + 0.0370·35-s + 0.903·37-s − 0.591·39-s − 0.674·41-s − 0.367·43-s − 0.115·45-s + 0.574·47-s − 0.988·49-s − 0.0684·51-s − 1.81·53-s + 0.711·57-s − 0.347·59-s − 0.434·61-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1452 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1452 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$1452$$    =    $$2^{2} \cdot 3 \cdot 11^{2}$$ Sign: $-1$ Analytic conductor: $$85.6707$$ Root analytic conductor: $$9.25585$$ Motivic weight: $$3$$ Rational: no Arithmetic: yes Character: Trivial Primitive: yes Self-dual: yes Analytic rank: $$1$$ Selberg data: $$(2,\ 1452,\ (\ :3/2),\ -1)$$

## 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$$F_p(T)$
bad2 $$1$$
3 $$1 - 3T$$
11 $$1$$
good5 $$1 + 3.88T + 125T^{2}$$
7 $$1 + 1.97T + 343T^{2}$$
13 $$1 + 48.0T + 2.19e3T^{2}$$
17 $$1 + 8.31T + 4.91e3T^{2}$$
19 $$1 - 102.T + 6.85e3T^{2}$$
23 $$1 - 108.T + 1.21e4T^{2}$$
29 $$1 + 88.1T + 2.43e4T^{2}$$
31 $$1 - 71.7T + 2.97e4T^{2}$$
37 $$1 - 203.T + 5.06e4T^{2}$$
41 $$1 + 177.T + 6.89e4T^{2}$$
43 $$1 + 103.T + 7.95e4T^{2}$$
47 $$1 - 185.T + 1.03e5T^{2}$$
53 $$1 + 698.T + 1.48e5T^{2}$$
59 $$1 + 157.T + 2.05e5T^{2}$$
61 $$1 + 207.T + 2.26e5T^{2}$$
67 $$1 - 404.T + 3.00e5T^{2}$$
71 $$1 + 22.5T + 3.57e5T^{2}$$
73 $$1 + 848.T + 3.89e5T^{2}$$
79 $$1 + 526.T + 4.93e5T^{2}$$
83 $$1 + 657.T + 5.71e5T^{2}$$
89 $$1 + 102.T + 7.04e5T^{2}$$
97 $$1 - 321.T + 9.12e5T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$