Properties

Label 2-168e2-1.1-c1-0-108
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 + 2·13-s + 6·17-s + 4·19-s + 6·23-s − 25-s − 4·31-s − 10·37-s + 2·41-s − 4·43-s − 4·47-s − 12·53-s + 4·55-s + 12·59-s + 6·61-s − 4·65-s − 4·67-s − 14·71-s + 2·73-s + 8·79-s − 16·83-s − 12·85-s − 6·89-s − 8·95-s + 18·97-s + 101-s + ⋯
L(s)  = 1  − 0.894·5-s − 0.603·11-s + 0.554·13-s + 1.45·17-s + 0.917·19-s + 1.25·23-s − 1/5·25-s − 0.718·31-s − 1.64·37-s + 0.312·41-s − 0.609·43-s − 0.583·47-s − 1.64·53-s + 0.539·55-s + 1.56·59-s + 0.768·61-s − 0.496·65-s − 0.488·67-s − 1.66·71-s + 0.234·73-s + 0.900·79-s − 1.75·83-s − 1.30·85-s − 0.635·89-s − 0.820·95-s + 1.82·97-s + 0.0995·101-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 \( 1 + 2 T + p T^{2} \)
11 \( 1 + 2 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 + 12 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 14 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 16 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 18 T + p T^{2} \)
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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

−15.68963410479808, −14.90483759940796, −14.51724602107292, −13.92438741452931, −13.28816191153300, −12.81722556350425, −12.21511082332753, −11.75282411334269, −11.19816446190910, −10.77259808315959, −9.974984155670035, −9.683329462620503, −8.710074328719777, −8.445639163906607, −7.590137027718273, −7.428656761241392, −6.722763758199981, −5.834680964646987, −5.305019682794390, −4.825976283234983, −3.874972738271155, −3.352275230224781, −2.946673919171185, −1.746817387467818, −1.003818466303674, 0, 1.003818466303674, 1.746817387467818, 2.946673919171185, 3.352275230224781, 3.874972738271155, 4.825976283234983, 5.305019682794390, 5.834680964646987, 6.722763758199981, 7.428656761241392, 7.590137027718273, 8.445639163906607, 8.710074328719777, 9.683329462620503, 9.974984155670035, 10.77259808315959, 11.19816446190910, 11.75282411334269, 12.21511082332753, 12.81722556350425, 13.28816191153300, 13.92438741452931, 14.51724602107292, 14.90483759940796, 15.68963410479808

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