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Properties

Label 2-168e2-1.1-c1-0-84
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 − 4·11-s + 2·13-s − 6·17-s + 8·19-s − 25-s + 6·29-s − 8·31-s + 2·37-s + 2·41-s + 4·43-s − 8·47-s + 6·53-s + 8·55-s − 6·61-s − 4·65-s + 4·67-s + 8·71-s − 10·73-s + 16·79-s − 8·83-s + 12·85-s − 6·89-s − 16·95-s + 6·97-s + 101-s + 103-s + ⋯
L(s)  = 1  − 0.894·5-s − 1.20·11-s + 0.554·13-s − 1.45·17-s + 1.83·19-s − 1/5·25-s + 1.11·29-s − 1.43·31-s + 0.328·37-s + 0.312·41-s + 0.609·43-s − 1.16·47-s + 0.824·53-s + 1.07·55-s − 0.768·61-s − 0.496·65-s + 0.488·67-s + 0.949·71-s − 1.17·73-s + 1.80·79-s − 0.878·83-s + 1.30·85-s − 0.635·89-s − 1.64·95-s + 0.609·97-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 \( 1 + 2 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 + 8 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
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

−15.61105119081659, −15.16333116787205, −14.37641080822871, −13.84540873812615, −13.31450247701690, −12.89282997824443, −12.25773904312798, −11.60636547928823, −11.25814886701025, −10.74631213996733, −10.15701774778506, −9.437986021075833, −8.933326433878998, −8.240494613937092, −7.790376980519505, −7.306625046815015, −6.698671856226858, −5.911024925700406, −5.287344736220134, −4.714598239794719, −4.031873406257992, −3.368077346186379, −2.754008840113126, −1.943008135238969, −0.8730684585929438, 0, 0.8730684585929438, 1.943008135238969, 2.754008840113126, 3.368077346186379, 4.031873406257992, 4.714598239794719, 5.287344736220134, 5.911024925700406, 6.698671856226858, 7.306625046815015, 7.790376980519505, 8.240494613937092, 8.933326433878998, 9.437986021075833, 10.15701774778506, 10.74631213996733, 11.25814886701025, 11.60636547928823, 12.25773904312798, 12.89282997824443, 13.31450247701690, 13.84540873812615, 14.37641080822871, 15.16333116787205, 15.61105119081659

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