# Properties

 Label 2-2e6-1.1-c1-0-0 Degree $2$ Conductor $64$ Sign $1$ Analytic cond. $0.511042$ Root an. cond. $0.714872$ Motivic weight $1$ Arithmetic yes Rational yes Primitive yes Self-dual yes Analytic rank $0$

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## Dirichlet series

 L(s)  = 1 + 2·5-s − 3·9-s − 6·13-s + 2·17-s − 25-s + 10·29-s + 2·37-s + 10·41-s − 6·45-s − 7·49-s − 14·53-s + 10·61-s − 12·65-s − 6·73-s + 9·81-s + 4·85-s + 10·89-s + 18·97-s + 2·101-s − 6·109-s − 14·113-s + 18·117-s + ⋯
 L(s)  = 1 + 0.894·5-s − 9-s − 1.66·13-s + 0.485·17-s − 1/5·25-s + 1.85·29-s + 0.328·37-s + 1.56·41-s − 0.894·45-s − 49-s − 1.92·53-s + 1.28·61-s − 1.48·65-s − 0.702·73-s + 81-s + 0.433·85-s + 1.05·89-s + 1.82·97-s + 0.199·101-s − 0.574·109-s − 1.31·113-s + 1.66·117-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$64$$    =    $$2^{6}$$ Sign: $1$ Analytic conductor: $$0.511042$$ Root analytic conductor: $$0.714872$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(2,\ 64,\ (\ :1/2),\ 1)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.9270373386$$ $$L(\frac12)$$ $$\approx$$ $$0.9270373386$$ $$L(\frac{3}{2})$$ 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 + p T^{2}$$
5 $$1 - 2 T + p T^{2}$$
7 $$1 + p T^{2}$$
11 $$1 + p T^{2}$$
13 $$1 + 6 T + p T^{2}$$
17 $$1 - 2 T + p T^{2}$$
19 $$1 + p T^{2}$$
23 $$1 + p T^{2}$$
29 $$1 - 10 T + p T^{2}$$
31 $$1 + p T^{2}$$
37 $$1 - 2 T + p T^{2}$$
41 $$1 - 10 T + p T^{2}$$
43 $$1 + p T^{2}$$
47 $$1 + p T^{2}$$
53 $$1 + 14 T + p T^{2}$$
59 $$1 + p T^{2}$$
61 $$1 - 10 T + p T^{2}$$
67 $$1 + p T^{2}$$
71 $$1 + p T^{2}$$
73 $$1 + 6 T + p T^{2}$$
79 $$1 + p T^{2}$$
83 $$1 + p T^{2}$$
89 $$1 - 10 T + p T^{2}$$
97 $$1 - 18 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

−14.59250078389593402269438368141, −14.09304813859309984769942580998, −12.72487277908244699547747964129, −11.66649219374340327203675385621, −10.23421229877843747435058592930, −9.317356556158666156286958675354, −7.86574055998535746161422058704, −6.27282080646876620831649653268, −5.01452297596172644651029655270, −2.63895104553179739845027310173, 2.63895104553179739845027310173, 5.01452297596172644651029655270, 6.27282080646876620831649653268, 7.86574055998535746161422058704, 9.317356556158666156286958675354, 10.23421229877843747435058592930, 11.66649219374340327203675385621, 12.72487277908244699547747964129, 14.09304813859309984769942580998, 14.59250078389593402269438368141