Properties

Label 2-168e2-1.1-c1-0-138
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 + 6·13-s − 4·17-s + 6·19-s + 4·23-s − 25-s − 6·29-s + 4·31-s + 6·37-s + 4·41-s − 12·43-s − 12·47-s + 6·53-s − 8·55-s − 6·59-s − 6·61-s − 12·65-s − 12·67-s − 8·71-s − 6·83-s + 8·85-s − 16·89-s − 12·95-s + 12·97-s + 101-s + 103-s + ⋯
L(s)  = 1  − 0.894·5-s + 1.20·11-s + 1.66·13-s − 0.970·17-s + 1.37·19-s + 0.834·23-s − 1/5·25-s − 1.11·29-s + 0.718·31-s + 0.986·37-s + 0.624·41-s − 1.82·43-s − 1.75·47-s + 0.824·53-s − 1.07·55-s − 0.781·59-s − 0.768·61-s − 1.48·65-s − 1.46·67-s − 0.949·71-s − 0.658·83-s + 0.867·85-s − 1.69·89-s − 1.23·95-s + 1.21·97-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: $\chi_{28224} (1, \cdot )$
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 \( 1 - 4 T + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 - 4 T + p T^{2} \)
43 \( 1 + 12 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 + 16 T + p T^{2} \)
97 \( 1 - 12 T + p T^{2} \)
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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

−15.34755846251322, −15.13562915279668, −14.49793006114185, −13.75604496012206, −13.38750495144183, −12.96698730368697, −12.06981113145555, −11.58866668229686, −11.32875758536763, −10.91793038241248, −9.992245390283334, −9.440525749971564, −8.831459408744054, −8.497221375828278, −7.719742100245167, −7.273730847793301, −6.470663637376625, −6.186788358241313, −5.335670685435215, −4.526560196132558, −4.014568836505442, −3.445869770715749, −2.901334472850476, −1.584322565553055, −1.168252463241035, 0, 1.168252463241035, 1.584322565553055, 2.901334472850476, 3.445869770715749, 4.014568836505442, 4.526560196132558, 5.335670685435215, 6.186788358241313, 6.470663637376625, 7.273730847793301, 7.719742100245167, 8.497221375828278, 8.831459408744054, 9.440525749971564, 9.992245390283334, 10.91793038241248, 11.32875758536763, 11.58866668229686, 12.06981113145555, 12.96698730368697, 13.38750495144183, 13.75604496012206, 14.49793006114185, 15.13562915279668, 15.34755846251322

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