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Properties

Label 2-168e2-1.1-c1-0-129
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 − 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: 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 \( 1 - T + p T^{2} \)
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

−15.55441895756006, −14.69824610529167, −14.60632906767425, −13.84504071343723, −13.42551001641051, −12.72966717113042, −12.36186501549864, −11.87840828834652, −10.93759541274628, −10.74122978824393, −10.08387819415434, −9.465614233962379, −9.076981229279384, −8.360752887566454, −7.625322867237592, −7.383861134453706, −6.524897384804281, −5.852684686782787, −5.411717745936974, −4.901718644656622, −3.810666816795462, −3.532979145875779, −2.573603961362115, −1.890923064123626, −1.120441792722620, 0, 1.120441792722620, 1.890923064123626, 2.573603961362115, 3.532979145875779, 3.810666816795462, 4.901718644656622, 5.411717745936974, 5.852684686782787, 6.524897384804281, 7.383861134453706, 7.625322867237592, 8.360752887566454, 9.076981229279384, 9.465614233962379, 10.08387819415434, 10.74122978824393, 10.93759541274628, 11.87840828834652, 12.36186501549864, 12.72966717113042, 13.42551001641051, 13.84504071343723, 14.60632906767425, 14.69824610529167, 15.55441895756006

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