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Properties

Label 2-96192-1.1-c1-0-24
Degree $2$
Conductor $96192$
Sign $-1$
Analytic cond. $768.096$
Root an. cond. $27.7145$
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 + 4·13-s + 4·19-s + 4·23-s − 25-s − 6·29-s − 8·31-s − 8·37-s + 4·43-s − 8·47-s − 7·49-s + 6·53-s + 8·55-s − 14·59-s + 2·61-s + 8·65-s − 8·67-s + 6·73-s + 14·79-s − 10·83-s + 6·89-s + 8·95-s − 18·97-s + 101-s + 103-s + 107-s + ⋯
L(s)  = 1  + 0.894·5-s + 1.20·11-s + 1.10·13-s + 0.917·19-s + 0.834·23-s − 1/5·25-s − 1.11·29-s − 1.43·31-s − 1.31·37-s + 0.609·43-s − 1.16·47-s − 49-s + 0.824·53-s + 1.07·55-s − 1.82·59-s + 0.256·61-s + 0.992·65-s − 0.977·67-s + 0.702·73-s + 1.57·79-s − 1.09·83-s + 0.635·89-s + 0.820·95-s − 1.82·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 96192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 96192 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(96192\)    =    \(2^{6} \cdot 3^{2} \cdot 167\)
Sign: $-1$
Analytic conductor: \(768.096\)
Root analytic conductor: \(27.7145\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 96192,\ (\ :1/2),\ -1)\)

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 \( 1 \)
167 \( 1 - T \)
good5 \( 1 - 2 T + p T^{2} \)
7 \( 1 + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 14 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 - 14 T + p T^{2} \)
83 \( 1 + 10 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 18 T + p T^{2} \)
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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

−13.87396613077718, −13.64833038104926, −13.18920970622781, −12.57753763886748, −12.16091823815143, −11.39185213757453, −11.20751332006093, −10.64767007394460, −10.03068092412802, −9.392586579244222, −9.178319336015903, −8.807215237385446, −8.002481685765527, −7.486628392646652, −6.813641167092941, −6.480582312846498, −5.802175796867556, −5.476059801815789, −4.870775775027197, −4.034475436008700, −3.537133809576699, −3.131130160120669, −2.113034406471539, −1.534764255095732, −1.191139755692015, 0, 1.191139755692015, 1.534764255095732, 2.113034406471539, 3.131130160120669, 3.537133809576699, 4.034475436008700, 4.870775775027197, 5.476059801815789, 5.802175796867556, 6.480582312846498, 6.813641167092941, 7.486628392646652, 8.002481685765527, 8.807215237385446, 9.178319336015903, 9.392586579244222, 10.03068092412802, 10.64767007394460, 11.20751332006093, 11.39185213757453, 12.16091823815143, 12.57753763886748, 13.18920970622781, 13.64833038104926, 13.87396613077718

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