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Properties

Label 2-168e2-1.1-c1-0-127
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  + 4·11-s − 8·23-s − 5·25-s + 2·29-s + 6·37-s + 12·43-s − 10·53-s − 4·67-s − 16·71-s + 8·79-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯
L(s)  = 1  + 1.20·11-s − 1.66·23-s − 25-s + 0.371·29-s + 0.986·37-s + 1.82·43-s − 1.37·53-s − 0.488·67-s − 1.89·71-s + 0.900·79-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: 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 \( 1 + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - 12 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 16 T + p T^{2} \)
73 \( 1 + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 + p T^{2} \)
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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

−15.54807293408806, −14.77884831371177, −14.47422798244651, −13.88667526414848, −13.50383505144056, −12.76361498433701, −12.04642761098775, −11.95806254908191, −11.21398967826502, −10.67298368001286, −9.997015570092439, −9.430383977690555, −9.136609498471160, −8.241465489456920, −7.860820698644434, −7.222809916218860, −6.437271322318744, −6.050671549307894, −5.518750446294870, −4.395791086145914, −4.212103789657070, −3.461760767959582, −2.603063468016341, −1.845305487402979, −1.108801751878753, 0, 1.108801751878753, 1.845305487402979, 2.603063468016341, 3.461760767959582, 4.212103789657070, 4.395791086145914, 5.518750446294870, 6.050671549307894, 6.437271322318744, 7.222809916218860, 7.860820698644434, 8.241465489456920, 9.136609498471160, 9.430383977690555, 9.997015570092439, 10.67298368001286, 11.21398967826502, 11.95806254908191, 12.04642761098775, 12.76361498433701, 13.50383505144056, 13.88667526414848, 14.47422798244651, 14.77884831371177, 15.54807293408806

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