Properties

Label 2-168e2-1.1-c1-0-128
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  + 4·11-s + 2·13-s − 4·17-s − 4·23-s − 5·25-s + 4·29-s − 8·31-s + 2·37-s + 4·41-s + 8·43-s + 8·47-s − 4·53-s − 8·59-s − 2·61-s + 8·67-s − 12·71-s − 6·73-s + 8·79-s − 16·83-s + 12·89-s + 2·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯
L(s)  = 1  + 1.20·11-s + 0.554·13-s − 0.970·17-s − 0.834·23-s − 25-s + 0.742·29-s − 1.43·31-s + 0.328·37-s + 0.624·41-s + 1.21·43-s + 1.16·47-s − 0.549·53-s − 1.04·59-s − 0.256·61-s + 0.977·67-s − 1.42·71-s − 0.702·73-s + 0.900·79-s − 1.75·83-s + 1.27·89-s + 0.203·97-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 \( 1 - 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 4 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 4 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 4 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 16 T + p T^{2} \)
89 \( 1 - 12 T + p T^{2} \)
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

−15.62210979692237, −14.88309589557096, −14.35920628436705, −13.93659001556333, −13.43908532878070, −12.80560861751549, −12.19883987034277, −11.81135595982775, −11.04272135411728, −10.87493057836630, −10.01062236533033, −9.408798794546004, −8.982613096291065, −8.502833866477195, −7.674226476722352, −7.270467917611914, −6.410858480859927, −6.103123357642331, −5.492556805322937, −4.458185892964228, −4.110761228942796, −3.527382974166142, −2.568162457837218, −1.856290665619364, −1.110582579913418, 0, 1.110582579913418, 1.856290665619364, 2.568162457837218, 3.527382974166142, 4.110761228942796, 4.458185892964228, 5.492556805322937, 6.103123357642331, 6.410858480859927, 7.270467917611914, 7.674226476722352, 8.502833866477195, 8.982613096291065, 9.408798794546004, 10.01062236533033, 10.87493057836630, 11.04272135411728, 11.81135595982775, 12.19883987034277, 12.80560861751549, 13.43908532878070, 13.93659001556333, 14.35920628436705, 14.88309589557096, 15.62210979692237

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