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Properties

Label 2-168e2-1.1-c1-0-135
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  + 5·13-s + 19-s − 5·25-s + 11·31-s − 11·37-s − 13·43-s + 14·61-s + 5·67-s − 17·73-s − 17·79-s − 14·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯
L(s)  = 1  + 1.38·13-s + 0.229·19-s − 25-s + 1.97·31-s − 1.80·37-s − 1.98·43-s + 1.79·61-s + 0.610·67-s − 1.98·73-s − 1.91·79-s − 1.42·97-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: 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 \( 1 + p T^{2} \)
13 \( 1 - 5 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 - 11 T + p T^{2} \)
37 \( 1 + 11 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 13 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 - 5 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 17 T + p T^{2} \)
79 \( 1 + 17 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−15.63116457726721, −15.07070374740997, −14.27670218128137, −13.85914800537735, −13.38055037404078, −12.98585734445798, −12.17321515758908, −11.58473717114824, −11.43935102939751, −10.48825466218619, −10.14316782393955, −9.628009527022309, −8.701476231005860, −8.468122271932340, −7.942768256716897, −7.029541210918455, −6.649030930612519, −5.948342044598071, −5.432015739706654, −4.695237629992141, −3.967180047218276, −3.431074276548241, −2.724100396444250, −1.752624007921889, −1.146924078325426, 0, 1.146924078325426, 1.752624007921889, 2.724100396444250, 3.431074276548241, 3.967180047218276, 4.695237629992141, 5.432015739706654, 5.948342044598071, 6.649030930612519, 7.029541210918455, 7.942768256716897, 8.468122271932340, 8.701476231005860, 9.628009527022309, 10.14316782393955, 10.48825466218619, 11.43935102939751, 11.58473717114824, 12.17321515758908, 12.98585734445798, 13.38055037404078, 13.85914800537735, 14.27670218128137, 15.07070374740997, 15.63116457726721

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