Properties

Label 2-168e2-1.1-c1-0-117
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  + 5-s − 5·11-s + 2·13-s + 6·17-s − 2·19-s − 6·23-s − 4·25-s − 3·29-s + 5·31-s + 2·37-s + 8·41-s − 4·43-s − 4·47-s + 9·53-s − 5·55-s − 3·59-s − 12·61-s + 2·65-s + 2·67-s − 8·71-s + 14·73-s − 79-s + 17·83-s + 6·85-s + 18·89-s − 2·95-s − 3·97-s + ⋯
L(s)  = 1  + 0.447·5-s − 1.50·11-s + 0.554·13-s + 1.45·17-s − 0.458·19-s − 1.25·23-s − 4/5·25-s − 0.557·29-s + 0.898·31-s + 0.328·37-s + 1.24·41-s − 0.609·43-s − 0.583·47-s + 1.23·53-s − 0.674·55-s − 0.390·59-s − 1.53·61-s + 0.248·65-s + 0.244·67-s − 0.949·71-s + 1.63·73-s − 0.112·79-s + 1.86·83-s + 0.650·85-s + 1.90·89-s − 0.205·95-s − 0.304·97-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 \( 1 - T + p T^{2} \)
11 \( 1 + 5 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 3 T + p T^{2} \)
31 \( 1 - 5 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 8 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 - 9 T + p T^{2} \)
59 \( 1 + 3 T + p T^{2} \)
61 \( 1 + 12 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 + T + p T^{2} \)
83 \( 1 - 17 T + p T^{2} \)
89 \( 1 - 18 T + p T^{2} \)
97 \( 1 + 3 T + p T^{2} \)
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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

−15.65208692277993, −14.87622835032886, −14.46135681618393, −13.72457946028930, −13.41953384875814, −12.96168726462262, −12.12375816982625, −11.98142030733680, −11.03349745659583, −10.57232859220679, −10.08228498404630, −9.666091394913413, −8.983148720872494, −8.128938866724337, −7.872402664627111, −7.421199988173928, −6.269763251194026, −6.066212422645715, −5.385682960829310, −4.849011035736333, −3.969503303500739, −3.381465116794934, −2.546128863744464, −2.003843543040834, −1.044682140069783, 0, 1.044682140069783, 2.003843543040834, 2.546128863744464, 3.381465116794934, 3.969503303500739, 4.849011035736333, 5.385682960829310, 6.066212422645715, 6.269763251194026, 7.421199988173928, 7.872402664627111, 8.128938866724337, 8.983148720872494, 9.666091394913413, 10.08228498404630, 10.57232859220679, 11.03349745659583, 11.98142030733680, 12.12375816982625, 12.96168726462262, 13.41953384875814, 13.72457946028930, 14.46135681618393, 14.87622835032886, 15.65208692277993

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