Properties

Label 2-168e2-1.1-c1-0-0
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  − 5-s + 11-s − 8·17-s − 4·19-s − 4·23-s − 4·25-s − 5·29-s − 7·31-s − 8·37-s + 4·41-s − 10·43-s + 6·47-s − 53-s − 55-s − 9·59-s − 2·61-s − 2·67-s − 6·71-s − 2·73-s − 9·79-s + 3·83-s + 8·85-s − 6·89-s + 4·95-s + 97-s + 101-s + 103-s + ⋯
L(s)  = 1  − 0.447·5-s + 0.301·11-s − 1.94·17-s − 0.917·19-s − 0.834·23-s − 4/5·25-s − 0.928·29-s − 1.25·31-s − 1.31·37-s + 0.624·41-s − 1.52·43-s + 0.875·47-s − 0.137·53-s − 0.134·55-s − 1.17·59-s − 0.256·61-s − 0.244·67-s − 0.712·71-s − 0.234·73-s − 1.01·79-s + 0.329·83-s + 0.867·85-s − 0.635·89-s + 0.410·95-s + 0.101·97-s + 0.0995·101-s + 0.0985·103-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

\(L(1)\) \(\approx\) \(0.3023174818\)
\(L(\frac12)\) \(\approx\) \(0.3023174818\)
\(L(\frac{3}{2})\) 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 \( 1 + T + p T^{2} \)
11 \( 1 - T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 + 8 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 5 T + p T^{2} \)
31 \( 1 + 7 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 - 4 T + p T^{2} \)
43 \( 1 + 10 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 + T + p T^{2} \)
59 \( 1 + 9 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 9 T + p T^{2} \)
83 \( 1 - 3 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - T + p T^{2} \)
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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

−15.26796149669517, −14.80082526664060, −14.12632690750885, −13.59927790450871, −13.07943405612665, −12.60473739557571, −11.92387325049877, −11.48462348148807, −10.89911183039987, −10.52562332509967, −9.754925231182854, −9.077999955522964, −8.738241408353613, −8.135625725971625, −7.384148399372192, −6.983009949788786, −6.248587024322730, −5.792269478855358, −4.921205906880999, −4.235610139840471, −3.895064621155727, −3.086599076200652, −2.014463448289110, −1.787217205468160, −0.1986243258091365, 0.1986243258091365, 1.787217205468160, 2.014463448289110, 3.086599076200652, 3.895064621155727, 4.235610139840471, 4.921205906880999, 5.792269478855358, 6.248587024322730, 6.983009949788786, 7.384148399372192, 8.135625725971625, 8.738241408353613, 9.077999955522964, 9.754925231182854, 10.52562332509967, 10.89911183039987, 11.48462348148807, 11.92387325049877, 12.60473739557571, 13.07943405612665, 13.59927790450871, 14.12632690750885, 14.80082526664060, 15.26796149669517

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