# Properties

 Label 2-2e7-1.1-c1-0-3 Degree $2$ Conductor $128$ Sign $-1$ Analytic cond. $1.02208$ Root an. cond. $1.01098$ Motivic weight $1$ Arithmetic yes Rational yes Primitive yes Self-dual yes Analytic rank $1$

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## Dirichlet series

 L(s)  = 1 − 2·3-s − 2·5-s − 4·7-s + 9-s + 2·11-s − 2·13-s + 4·15-s − 2·17-s − 2·19-s + 8·21-s + 4·23-s − 25-s + 4·27-s + 6·29-s − 4·33-s + 8·35-s − 10·37-s + 4·39-s − 6·41-s − 6·43-s − 2·45-s − 8·47-s + 9·49-s + 4·51-s + 6·53-s − 4·55-s + 4·57-s + ⋯
 L(s)  = 1 − 1.15·3-s − 0.894·5-s − 1.51·7-s + 1/3·9-s + 0.603·11-s − 0.554·13-s + 1.03·15-s − 0.485·17-s − 0.458·19-s + 1.74·21-s + 0.834·23-s − 1/5·25-s + 0.769·27-s + 1.11·29-s − 0.696·33-s + 1.35·35-s − 1.64·37-s + 0.640·39-s − 0.937·41-s − 0.914·43-s − 0.298·45-s − 1.16·47-s + 9/7·49-s + 0.560·51-s + 0.824·53-s − 0.539·55-s + 0.529·57-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$128$$    =    $$2^{7}$$ Sign: $-1$ Analytic conductor: $$1.02208$$ Root analytic conductor: $$1.01098$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: yes Self-dual: yes Analytic rank: $$1$$ Selberg data: $$(2,\ 128,\ (\ :1/2),\ -1)$$

## 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$$
good3 $$1 + 2 T + p T^{2}$$
5 $$1 + 2 T + p T^{2}$$
7 $$1 + 4 T + p T^{2}$$
11 $$1 - 2 T + p T^{2}$$
13 $$1 + 2 T + p T^{2}$$
17 $$1 + 2 T + p T^{2}$$
19 $$1 + 2 T + p T^{2}$$
23 $$1 - 4 T + p T^{2}$$
29 $$1 - 6 T + p T^{2}$$
31 $$1 + p T^{2}$$
37 $$1 + 10 T + p T^{2}$$
41 $$1 + 6 T + p T^{2}$$
43 $$1 + 6 T + p T^{2}$$
47 $$1 + 8 T + p T^{2}$$
53 $$1 - 6 T + p T^{2}$$
59 $$1 + 14 T + p T^{2}$$
61 $$1 + 2 T + p T^{2}$$
67 $$1 + 10 T + p T^{2}$$
71 $$1 - 12 T + p T^{2}$$
73 $$1 - 14 T + p T^{2}$$
79 $$1 + 8 T + p T^{2}$$
83 $$1 - 6 T + p T^{2}$$
89 $$1 + 2 T + p T^{2}$$
97 $$1 + 2 T + p T^{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

−12.44338350662649173502905410996, −11.96377224112661393265117422678, −10.90944990278511493765148592647, −9.886348706897054809183790479328, −8.635939205168057076823393046651, −6.97464122003923989664965494039, −6.33050260025483473718379487161, −4.84373431553687523132526670248, −3.36730269661969347724618284285, 0, 3.36730269661969347724618284285, 4.84373431553687523132526670248, 6.33050260025483473718379487161, 6.97464122003923989664965494039, 8.635939205168057076823393046651, 9.886348706897054809183790479328, 10.90944990278511493765148592647, 11.96377224112661393265117422678, 12.44338350662649173502905410996