# Properties

 Label 2-76-1.1-c3-0-3 Degree $2$ Conductor $76$ Sign $-1$ Analytic cond. $4.48414$ Root an. cond. $2.11758$ Motivic weight $3$ Arithmetic yes Rational no Primitive yes Self-dual yes Analytic rank $1$

# Related objects

## Dirichlet series

 L(s)  = 1 − 5.37·3-s + 11.8·5-s − 26.4·7-s + 1.86·9-s − 49.8·11-s − 49.0·13-s − 63.7·15-s + 17.2·17-s − 19·19-s + 142.·21-s + 166.·23-s + 15.6·25-s + 135.·27-s − 109.·29-s − 273.·31-s + 267.·33-s − 314.·35-s + 167.·37-s + 263.·39-s + 15.1·41-s + 413.·43-s + 22.0·45-s − 161.·47-s + 358.·49-s − 92.5·51-s − 490.·53-s − 591.·55-s + ⋯
 L(s)  = 1 − 1.03·3-s + 1.06·5-s − 1.43·7-s + 0.0689·9-s − 1.36·11-s − 1.04·13-s − 1.09·15-s + 0.245·17-s − 0.229·19-s + 1.47·21-s + 1.51·23-s + 0.125·25-s + 0.962·27-s − 0.699·29-s − 1.58·31-s + 1.41·33-s − 1.51·35-s + 0.742·37-s + 1.08·39-s + 0.0577·41-s + 1.46·43-s + 0.0731·45-s − 0.501·47-s + 1.04·49-s − 0.254·51-s − 1.27·53-s − 1.44·55-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$76$$    =    $$2^{2} \cdot 19$$ Sign: $-1$ Analytic conductor: $$4.48414$$ Root analytic conductor: $$2.11758$$ Motivic weight: $$3$$ Rational: no Arithmetic: yes Character: $\chi_{76} (1, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$1$$ Selberg data: $$(2,\ 76,\ (\ :3/2),\ -1)$$

## 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$$
19 $$1 + 19T$$
good3 $$1 + 5.37T + 27T^{2}$$
5 $$1 - 11.8T + 125T^{2}$$
7 $$1 + 26.4T + 343T^{2}$$
11 $$1 + 49.8T + 1.33e3T^{2}$$
13 $$1 + 49.0T + 2.19e3T^{2}$$
17 $$1 - 17.2T + 4.91e3T^{2}$$
23 $$1 - 166.T + 1.21e4T^{2}$$
29 $$1 + 109.T + 2.43e4T^{2}$$
31 $$1 + 273.T + 2.97e4T^{2}$$
37 $$1 - 167.T + 5.06e4T^{2}$$
41 $$1 - 15.1T + 6.89e4T^{2}$$
43 $$1 - 413.T + 7.95e4T^{2}$$
47 $$1 + 161.T + 1.03e5T^{2}$$
53 $$1 + 490.T + 1.48e5T^{2}$$
59 $$1 + 335.T + 2.05e5T^{2}$$
61 $$1 - 725.T + 2.26e5T^{2}$$
67 $$1 - 497.T + 3.00e5T^{2}$$
71 $$1 + 798.T + 3.57e5T^{2}$$
73 $$1 + 311.T + 3.89e5T^{2}$$
79 $$1 + 665.T + 4.93e5T^{2}$$
83 $$1 + 372.T + 5.71e5T^{2}$$
89 $$1 + 673.T + 7.04e5T^{2}$$
97 $$1 + 960.T + 9.12e5T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$