Properties

Label 2-47096-1.1-c1-0-8
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 + 2·5-s − 7-s + 9-s + 6·13-s − 4·15-s − 4·17-s − 2·19-s + 2·21-s − 25-s + 4·27-s + 10·31-s − 2·35-s − 2·37-s − 12·39-s − 12·43-s + 2·45-s − 6·47-s + 49-s + 8·51-s + 6·53-s + 4·57-s − 4·59-s − 4·61-s − 63-s + 12·65-s + 4·67-s + ⋯
L(s)  = 1  − 1.15·3-s + 0.894·5-s − 0.377·7-s + 1/3·9-s + 1.66·13-s − 1.03·15-s − 0.970·17-s − 0.458·19-s + 0.436·21-s − 1/5·25-s + 0.769·27-s + 1.79·31-s − 0.338·35-s − 0.328·37-s − 1.92·39-s − 1.82·43-s + 0.298·45-s − 0.875·47-s + 1/7·49-s + 1.12·51-s + 0.824·53-s + 0.529·57-s − 0.520·59-s − 0.512·61-s − 0.125·63-s + 1.48·65-s + 0.488·67-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 \( 1 - 2 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 + p T^{2} \)
31 \( 1 - 10 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 12 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 4 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 - 4 T + p T^{2} \)
97 \( 1 - 4 T + p T^{2} \)
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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

−15.08384450148809, −14.00948586824951, −13.81704798165432, −13.21913494782329, −12.95596866904372, −12.19635868632897, −11.55660787873176, −11.38693917409908, −10.63658513815121, −10.30165303193217, −9.833999072066445, −9.053371345478145, −8.583199087531327, −8.168754265379330, −7.153582302861251, −6.483055916421887, −6.169215607803521, −6.024653513086353, −5.056462638713431, −4.747433874750181, −3.873325527519389, −3.235646843158298, −2.385248488072640, −1.645418373619752, −0.9144165330146104, 0, 0.9144165330146104, 1.645418373619752, 2.385248488072640, 3.235646843158298, 3.873325527519389, 4.747433874750181, 5.056462638713431, 6.024653513086353, 6.169215607803521, 6.483055916421887, 7.153582302861251, 8.168754265379330, 8.583199087531327, 9.053371345478145, 9.833999072066445, 10.30165303193217, 10.63658513815121, 11.38693917409908, 11.55660787873176, 12.19635868632897, 12.95596866904372, 13.21913494782329, 13.81704798165432, 14.00948586824951, 15.08384450148809

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