# Properties

 Label 2-4334-1.1-c1-0-93 Degree $2$ Conductor $4334$ Sign $-1$ Analytic cond. $34.6071$ Root an. cond. $5.88278$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual yes Analytic rank $1$

# Related objects

## Dirichlet series

 L(s)  = 1 − 2-s − 2.47·3-s + 4-s + 1.17·5-s + 2.47·6-s + 0.539·7-s − 8-s + 3.13·9-s − 1.17·10-s + 11-s − 2.47·12-s + 3.77·13-s − 0.539·14-s − 2.90·15-s + 16-s − 5.84·17-s − 3.13·18-s − 0.884·19-s + 1.17·20-s − 1.33·21-s − 22-s + 5.41·23-s + 2.47·24-s − 3.62·25-s − 3.77·26-s − 0.339·27-s + 0.539·28-s + ⋯
 L(s)  = 1 − 0.707·2-s − 1.43·3-s + 0.5·4-s + 0.525·5-s + 1.01·6-s + 0.204·7-s − 0.353·8-s + 1.04·9-s − 0.371·10-s + 0.301·11-s − 0.715·12-s + 1.04·13-s − 0.144·14-s − 0.751·15-s + 0.250·16-s − 1.41·17-s − 0.739·18-s − 0.202·19-s + 0.262·20-s − 0.291·21-s − 0.213·22-s + 1.12·23-s + 0.505·24-s − 0.724·25-s − 0.739·26-s − 0.0653·27-s + 0.102·28-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 4334 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 4334 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$4334$$    =    $$2 \cdot 11 \cdot 197$$ Sign: $-1$ Analytic conductor: $$34.6071$$ Root analytic conductor: $$5.88278$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: Trivial Primitive: yes Self-dual: yes Analytic rank: $$1$$ Selberg data: $$(2,\ 4334,\ (\ :1/2),\ -1)$$

## Particular Values

 $$L(1)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + T$$
11 $$1 - T$$
197 $$1 - T$$
good3 $$1 + 2.47T + 3T^{2}$$
5 $$1 - 1.17T + 5T^{2}$$
7 $$1 - 0.539T + 7T^{2}$$
13 $$1 - 3.77T + 13T^{2}$$
17 $$1 + 5.84T + 17T^{2}$$
19 $$1 + 0.884T + 19T^{2}$$
23 $$1 - 5.41T + 23T^{2}$$
29 $$1 + 1.87T + 29T^{2}$$
31 $$1 - 1.14T + 31T^{2}$$
37 $$1 - 2.52T + 37T^{2}$$
41 $$1 + 1.15T + 41T^{2}$$
43 $$1 + 10.8T + 43T^{2}$$
47 $$1 - 0.607T + 47T^{2}$$
53 $$1 - 2.28T + 53T^{2}$$
59 $$1 + 7.45T + 59T^{2}$$
61 $$1 + 8.34T + 61T^{2}$$
67 $$1 + 14.3T + 67T^{2}$$
71 $$1 + 6.18T + 71T^{2}$$
73 $$1 - 12.6T + 73T^{2}$$
79 $$1 - 15.3T + 79T^{2}$$
83 $$1 - 13.5T + 83T^{2}$$
89 $$1 + 4.22T + 89T^{2}$$
97 $$1 + 14.4T + 97T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$