Properties

Label 2-168e2-1.1-c1-0-125
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  + 4·13-s + 4·17-s − 4·19-s − 4·23-s − 5·25-s + 2·29-s − 8·31-s + 6·37-s + 12·41-s − 4·43-s − 8·47-s + 6·53-s − 12·59-s + 4·61-s + 4·67-s + 12·71-s − 8·73-s − 16·79-s + 4·83-s + 4·89-s − 16·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯
L(s)  = 1  + 1.10·13-s + 0.970·17-s − 0.917·19-s − 0.834·23-s − 25-s + 0.371·29-s − 1.43·31-s + 0.986·37-s + 1.87·41-s − 0.609·43-s − 1.16·47-s + 0.824·53-s − 1.56·59-s + 0.512·61-s + 0.488·67-s + 1.42·71-s − 0.936·73-s − 1.80·79-s + 0.439·83-s + 0.423·89-s − 1.62·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-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 \( 1 + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 - 12 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 4 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 8 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 4 T + p T^{2} \)
97 \( 1 + 16 T + p T^{2} \)
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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

−15.59017040667604, −14.87227676531472, −14.38415596236912, −14.00002531476371, −13.25698416527823, −12.87216069157611, −12.35198631588674, −11.66890221663568, −11.15735473434954, −10.74669808684759, −9.949014046397195, −9.666902783377643, −8.837401112516930, −8.405912445292860, −7.737899682261854, −7.366924063607934, −6.324123640429497, −6.081704716777286, −5.489208854565209, −4.638962171687224, −3.915299360433234, −3.558700825271299, −2.608892412063164, −1.850720435860068, −1.094704152881073, 0, 1.094704152881073, 1.850720435860068, 2.608892412063164, 3.558700825271299, 3.915299360433234, 4.638962171687224, 5.489208854565209, 6.081704716777286, 6.324123640429497, 7.366924063607934, 7.737899682261854, 8.405912445292860, 8.837401112516930, 9.666902783377643, 9.949014046397195, 10.74669808684759, 11.15735473434954, 11.66890221663568, 12.35198631588674, 12.87216069157611, 13.25698416527823, 14.00002531476371, 14.38415596236912, 14.87227676531472, 15.59017040667604

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