Properties

Label 2-168e2-1.1-c1-0-31
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·5-s − 4·11-s − 2·13-s + 2·17-s + 4·19-s − 8·23-s − 25-s + 6·29-s + 8·31-s − 6·37-s − 6·41-s + 4·43-s − 2·53-s − 8·55-s + 4·59-s − 2·61-s − 4·65-s − 4·67-s + 8·71-s − 10·73-s + 8·79-s − 4·83-s + 4·85-s − 6·89-s + 8·95-s − 2·97-s + 101-s + ⋯
L(s)  = 1  + 0.894·5-s − 1.20·11-s − 0.554·13-s + 0.485·17-s + 0.917·19-s − 1.66·23-s − 1/5·25-s + 1.11·29-s + 1.43·31-s − 0.986·37-s − 0.937·41-s + 0.609·43-s − 0.274·53-s − 1.07·55-s + 0.520·59-s − 0.256·61-s − 0.496·65-s − 0.488·67-s + 0.949·71-s − 1.17·73-s + 0.900·79-s − 0.439·83-s + 0.433·85-s − 0.635·89-s + 0.820·95-s − 0.203·97-s + 0.0995·101-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: \(0\)
Selberg data: \((2,\ 28224,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.080935415\)
\(L(\frac12)\) \(\approx\) \(2.080935415\)
\(L(\frac{3}{2})\) 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 \( 1 - 2 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
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

−15.41177576626597, −14.45978842020457, −14.08079225809731, −13.68533037083324, −13.23776361066785, −12.50941247530028, −11.97522992555728, −11.68948322644200, −10.66527083758008, −10.16114315097483, −9.990890391747834, −9.426193003635166, −8.537490230983719, −8.085867796445375, −7.551870739309040, −6.890786821047092, −6.099441340234828, −5.735484144634581, −5.043524502048112, −4.600249650037179, −3.604086989294294, −2.879622125114417, −2.304367975707785, −1.599838651620723, −0.5457161769945095, 0.5457161769945095, 1.599838651620723, 2.304367975707785, 2.879622125114417, 3.604086989294294, 4.600249650037179, 5.043524502048112, 5.735484144634581, 6.099441340234828, 6.890786821047092, 7.551870739309040, 8.085867796445375, 8.537490230983719, 9.426193003635166, 9.990890391747834, 10.16114315097483, 10.66527083758008, 11.68948322644200, 11.97522992555728, 12.50941247530028, 13.23776361066785, 13.68533037083324, 14.08079225809731, 14.45978842020457, 15.41177576626597

Graph of the $Z$-function along the critical line