Properties

Label 2-47096-1.1-c1-0-7
Degree $2$
Conductor $47096$
Sign $-1$
Analytic cond. $376.063$
Root an. cond. $19.3923$
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  − 2·3-s + 7-s + 9-s + 4·13-s − 2·17-s − 6·19-s − 2·21-s − 5·25-s + 4·27-s − 2·37-s − 8·39-s − 10·41-s + 8·43-s − 8·47-s + 49-s + 4·51-s + 6·53-s + 12·57-s + 10·59-s + 12·61-s + 63-s − 12·67-s + 8·71-s − 14·73-s + 10·75-s − 11·81-s + 2·83-s + ⋯
L(s)  = 1  − 1.15·3-s + 0.377·7-s + 1/3·9-s + 1.10·13-s − 0.485·17-s − 1.37·19-s − 0.436·21-s − 25-s + 0.769·27-s − 0.328·37-s − 1.28·39-s − 1.56·41-s + 1.21·43-s − 1.16·47-s + 1/7·49-s + 0.560·51-s + 0.824·53-s + 1.58·57-s + 1.30·59-s + 1.53·61-s + 0.125·63-s − 1.46·67-s + 0.949·71-s − 1.63·73-s + 1.15·75-s − 1.22·81-s + 0.219·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 47096 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 47096 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(47096\)    =    \(2^{3} \cdot 7 \cdot 29^{2}\)
Sign: $-1$
Analytic conductor: \(376.063\)
Root analytic conductor: \(19.3923\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{47096} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 47096,\ (\ :1/2),\ -1)\)

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 \( 1 \)
7 \( 1 - T \)
29 \( 1 \)
good3 \( 1 + 2 T + p T^{2} \)
5 \( 1 + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 10 T + p T^{2} \)
61 \( 1 - 12 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 2 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
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

−14.76959083491552, −14.49530565094828, −13.61911615093485, −13.30506931416188, −12.80171235050532, −12.10194952261605, −11.67582318356076, −11.26672465289566, −10.82077039964515, −10.30859234647449, −9.880461838998947, −8.811271660103754, −8.681160981520491, −8.066596073613741, −7.285564167549367, −6.634042545765372, −6.263173117133949, −5.714662700757663, −5.203822188646167, −4.507289658947354, −4.005270983695673, −3.323770913066300, −2.302030064657856, −1.716058284109685, −0.8082170297935037, 0, 0.8082170297935037, 1.716058284109685, 2.302030064657856, 3.323770913066300, 4.005270983695673, 4.507289658947354, 5.203822188646167, 5.714662700757663, 6.263173117133949, 6.634042545765372, 7.285564167549367, 8.066596073613741, 8.681160981520491, 8.811271660103754, 9.880461838998947, 10.30859234647449, 10.82077039964515, 11.26672465289566, 11.67582318356076, 12.10194952261605, 12.80171235050532, 13.30506931416188, 13.61911615093485, 14.49530565094828, 14.76959083491552

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