Properties

Label 2-168e2-1.1-c1-0-106
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  − 2·11-s − 2·13-s + 4·17-s − 4·19-s + 6·23-s − 5·25-s − 2·29-s + 6·37-s + 8·41-s + 8·43-s − 4·47-s − 6·53-s − 14·61-s − 4·67-s + 2·71-s + 2·73-s + 4·79-s − 12·83-s − 6·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯
L(s)  = 1  − 0.603·11-s − 0.554·13-s + 0.970·17-s − 0.917·19-s + 1.25·23-s − 25-s − 0.371·29-s + 0.986·37-s + 1.24·41-s + 1.21·43-s − 0.583·47-s − 0.824·53-s − 1.79·61-s − 0.488·67-s + 0.237·71-s + 0.234·73-s + 0.450·79-s − 1.31·83-s − 0.609·97-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 \( 1 + 2 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 - 8 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 2 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 + 6 T + p T^{2} \)
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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

−15.47040516023156, −14.90471146812614, −14.48684856966679, −13.95954538129584, −13.22759250122006, −12.82757936576742, −12.37708491385573, −11.78343349356056, −10.95290949885707, −10.85357672299928, −10.01041510208106, −9.511274632639436, −9.072632744527151, −8.246133674541910, −7.698061910687971, −7.393184285730511, −6.543763047282119, −5.888932163471303, −5.445670809351391, −4.624041664683009, −4.192256693844243, −3.234491348429692, −2.706025541652383, −1.941068633763612, −1.002588779467299, 0, 1.002588779467299, 1.941068633763612, 2.706025541652383, 3.234491348429692, 4.192256693844243, 4.624041664683009, 5.445670809351391, 5.888932163471303, 6.543763047282119, 7.393184285730511, 7.698061910687971, 8.246133674541910, 9.072632744527151, 9.511274632639436, 10.01041510208106, 10.85357672299928, 10.95290949885707, 11.78343349356056, 12.37708491385573, 12.82757936576742, 13.22759250122006, 13.95954538129584, 14.48684856966679, 14.90471146812614, 15.47040516023156

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