Properties

Label 2-168e2-1.1-c1-0-101
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 + 6·13-s + 2·17-s + 4·19-s + 8·23-s − 25-s − 2·29-s + 10·37-s − 6·41-s − 4·43-s + 6·53-s + 8·55-s + 4·59-s + 6·61-s + 12·65-s + 4·67-s + 8·71-s − 10·73-s − 4·83-s + 4·85-s − 6·89-s + 8·95-s + 14·97-s + 101-s + 103-s + 107-s + ⋯
L(s)  = 1  + 0.894·5-s + 1.20·11-s + 1.66·13-s + 0.485·17-s + 0.917·19-s + 1.66·23-s − 1/5·25-s − 0.371·29-s + 1.64·37-s − 0.937·41-s − 0.609·43-s + 0.824·53-s + 1.07·55-s + 0.520·59-s + 0.768·61-s + 1.48·65-s + 0.488·67-s + 0.949·71-s − 1.17·73-s − 0.439·83-s + 0.433·85-s − 0.635·89-s + 0.820·95-s + 1.42·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-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

\(L(1)\) \(\approx\) \(4.380018149\)
\(L(\frac12)\) \(\approx\) \(4.380018149\)
\(L(\frac{3}{2})\) 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 \( 1 - 6 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
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

−15.21388562812125, −14.51257185227895, −14.22911067343575, −13.45338717934197, −13.29381831402501, −12.72735657666329, −11.82949418965155, −11.42522122936979, −11.08243937382584, −10.19175655828558, −9.830267974455839, −9.122785198265599, −8.869988892338325, −8.168770677969363, −7.414240107640746, −6.721973763207813, −6.316133508638844, −5.661210930275668, −5.234074038591058, −4.311560087919987, −3.618245860922687, −3.149291213774677, −2.177541959622044, −1.274080513607606, −0.9882400813330538, 0.9882400813330538, 1.274080513607606, 2.177541959622044, 3.149291213774677, 3.618245860922687, 4.311560087919987, 5.234074038591058, 5.661210930275668, 6.316133508638844, 6.721973763207813, 7.414240107640746, 8.168770677969363, 8.869988892338325, 9.122785198265599, 9.830267974455839, 10.19175655828558, 11.08243937382584, 11.42522122936979, 11.82949418965155, 12.72735657666329, 13.29381831402501, 13.45338717934197, 14.22911067343575, 14.51257185227895, 15.21388562812125

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