# Properties

 Label 2-168e2-1.1-c1-0-135 Degree $2$ Conductor $28224$ Sign $-1$ Analytic cond. $225.369$ Root an. cond. $15.0123$ Motivic weight $1$ Arithmetic yes Rational yes Primitive yes Self-dual yes Analytic rank $1$

# Origins

## Dirichlet series

 L(s)  = 1 + 5·13-s + 19-s − 5·25-s + 11·31-s − 11·37-s − 13·43-s + 14·61-s + 5·67-s − 17·73-s − 17·79-s − 14·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯
 L(s)  = 1 + 1.38·13-s + 0.229·19-s − 25-s + 1.97·31-s − 1.80·37-s − 1.98·43-s + 1.79·61-s + 0.610·67-s − 1.98·73-s − 1.91·79-s − 1.42·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 28224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 28224 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$28224$$    =    $$2^{6} \cdot 3^{2} \cdot 7^{2}$$ Sign: $-1$ Analytic conductor: $$225.369$$ Root analytic conductor: $$15.0123$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: $\chi_{28224} (1, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$1$$ Selberg data: $$(2,\ 28224,\ (\ :1/2),\ -1)$$

## 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$$
3 $$1$$
7 $$1$$
good5 $$1 + p T^{2}$$
11 $$1 + p T^{2}$$
13 $$1 - 5 T + p T^{2}$$
17 $$1 + p T^{2}$$
19 $$1 - T + p T^{2}$$
23 $$1 + p T^{2}$$
29 $$1 + p T^{2}$$
31 $$1 - 11 T + p T^{2}$$
37 $$1 + 11 T + p T^{2}$$
41 $$1 + p T^{2}$$
43 $$1 + 13 T + p T^{2}$$
47 $$1 + p T^{2}$$
53 $$1 + p T^{2}$$
59 $$1 + p T^{2}$$
61 $$1 - 14 T + p T^{2}$$
67 $$1 - 5 T + p T^{2}$$
71 $$1 + p T^{2}$$
73 $$1 + 17 T + p T^{2}$$
79 $$1 + 17 T + p T^{2}$$
83 $$1 + p T^{2}$$
89 $$1 + p T^{2}$$
97 $$1 + 14 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

−15.63116457726721, −15.07070374740997, −14.27670218128137, −13.85914800537735, −13.38055037404078, −12.98585734445798, −12.17321515758908, −11.58473717114824, −11.43935102939751, −10.48825466218619, −10.14316782393955, −9.628009527022309, −8.701476231005860, −8.468122271932340, −7.942768256716897, −7.029541210918455, −6.649030930612519, −5.948342044598071, −5.432015739706654, −4.695237629992141, −3.967180047218276, −3.431074276548241, −2.724100396444250, −1.752624007921889, −1.146924078325426, 0, 1.146924078325426, 1.752624007921889, 2.724100396444250, 3.431074276548241, 3.967180047218276, 4.695237629992141, 5.432015739706654, 5.948342044598071, 6.649030930612519, 7.029541210918455, 7.942768256716897, 8.468122271932340, 8.701476231005860, 9.628009527022309, 10.14316782393955, 10.48825466218619, 11.43935102939751, 11.58473717114824, 12.17321515758908, 12.98585734445798, 13.38055037404078, 13.85914800537735, 14.27670218128137, 15.07070374740997, 15.63116457726721