Properties

Label 2-168e2-1.1-c1-0-142
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

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·5-s + 6·11-s − 6·13-s + 2·17-s − 4·19-s − 2·23-s − 25-s − 8·29-s + 4·31-s + 6·37-s − 10·41-s − 4·43-s − 4·47-s + 4·53-s + 12·55-s + 12·59-s − 2·61-s − 12·65-s + 12·67-s − 6·71-s + 2·73-s + 8·79-s + 4·85-s + 14·89-s − 8·95-s + 2·97-s + 101-s + ⋯
L(s)  = 1  + 0.894·5-s + 1.80·11-s − 1.66·13-s + 0.485·17-s − 0.917·19-s − 0.417·23-s − 1/5·25-s − 1.48·29-s + 0.718·31-s + 0.986·37-s − 1.56·41-s − 0.609·43-s − 0.583·47-s + 0.549·53-s + 1.61·55-s + 1.56·59-s − 0.256·61-s − 1.48·65-s + 1.46·67-s − 0.712·71-s + 0.234·73-s + 0.900·79-s + 0.433·85-s + 1.48·89-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: \(1\)
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 \( 1 - 2 T + p T^{2} \)
11 \( 1 - 6 T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 2 T + p T^{2} \)
29 \( 1 + 8 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 - 4 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
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

−15.15275742741257, −14.81334583791165, −14.57541086150906, −13.92591486938814, −13.38673749410195, −12.87552804188751, −12.12400904385238, −11.87119344728670, −11.31991427455652, −10.48684142443438, −9.843465758281898, −9.641825668492753, −9.145783552187186, −8.371729167960302, −7.827076493827706, −6.926477222256603, −6.693561305484115, −5.991441399967422, −5.395321048442340, −4.746395642317029, −4.019141137174079, −3.475078221058273, −2.389404428043695, −1.983416557945121, −1.183621701938340, 0, 1.183621701938340, 1.983416557945121, 2.389404428043695, 3.475078221058273, 4.019141137174079, 4.746395642317029, 5.395321048442340, 5.991441399967422, 6.693561305484115, 6.926477222256603, 7.827076493827706, 8.371729167960302, 9.145783552187186, 9.641825668492753, 9.843465758281898, 10.48684142443438, 11.31991427455652, 11.87119344728670, 12.12400904385238, 12.87552804188751, 13.38673749410195, 13.92591486938814, 14.57541086150906, 14.81334583791165, 15.15275742741257

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