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Properties

Label 2-168e2-1.1-c1-0-133
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 − 2·11-s + 13-s + 19-s − 25-s + 4·29-s − 9·31-s − 3·37-s − 10·41-s − 5·43-s − 6·47-s + 12·53-s − 4·55-s + 12·59-s + 10·61-s + 2·65-s + 5·67-s + 6·71-s + 3·73-s − 79-s − 6·83-s + 16·89-s + 2·95-s + 6·97-s + 101-s + 103-s + 107-s + ⋯
L(s)  = 1  + 0.894·5-s − 0.603·11-s + 0.277·13-s + 0.229·19-s − 1/5·25-s + 0.742·29-s − 1.61·31-s − 0.493·37-s − 1.56·41-s − 0.762·43-s − 0.875·47-s + 1.64·53-s − 0.539·55-s + 1.56·59-s + 1.28·61-s + 0.248·65-s + 0.610·67-s + 0.712·71-s + 0.351·73-s − 0.112·79-s − 0.658·83-s + 1.69·89-s + 0.205·95-s + 0.609·97-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: Trivial
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 \( 1 + 2 T + p T^{2} \)
13 \( 1 - T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 4 T + p T^{2} \)
31 \( 1 + 9 T + p T^{2} \)
37 \( 1 + 3 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 + 5 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 - 12 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 5 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 - 3 T + p T^{2} \)
79 \( 1 + T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 - 16 T + p T^{2} \)
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

−15.50680730500091, −14.80465420558698, −14.50422771334062, −13.71398476349116, −13.40323094058985, −12.99465305458869, −12.33160836112986, −11.67186630112009, −11.24352063363659, −10.40092685462328, −10.16268715637738, −9.618585986973871, −8.912880120928221, −8.429987470202972, −7.854757034964102, −7.014929130232083, −6.667582186615404, −5.895375554636038, −5.211639826468563, −5.087579819354276, −3.873794429336191, −3.458344329651308, −2.463414109605395, −1.988662908990574, −1.138154310741946, 0, 1.138154310741946, 1.988662908990574, 2.463414109605395, 3.458344329651308, 3.873794429336191, 5.087579819354276, 5.211639826468563, 5.895375554636038, 6.667582186615404, 7.014929130232083, 7.854757034964102, 8.429987470202972, 8.912880120928221, 9.618585986973871, 10.16268715637738, 10.40092685462328, 11.24352063363659, 11.67186630112009, 12.33160836112986, 12.99465305458869, 13.40323094058985, 13.71398476349116, 14.50422771334062, 14.80465420558698, 15.50680730500091

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