Properties

Label 2-47096-1.1-c1-0-9
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  + 2·5-s − 7-s − 3·9-s + 4·11-s + 2·13-s + 6·17-s − 8·19-s − 25-s − 8·31-s − 2·35-s + 2·37-s − 2·41-s + 4·43-s − 6·45-s + 8·47-s + 49-s + 6·53-s + 8·55-s + 6·61-s + 3·63-s + 4·65-s − 4·67-s − 8·71-s − 10·73-s − 4·77-s − 16·79-s + 9·81-s + ⋯
L(s)  = 1  + 0.894·5-s − 0.377·7-s − 9-s + 1.20·11-s + 0.554·13-s + 1.45·17-s − 1.83·19-s − 1/5·25-s − 1.43·31-s − 0.338·35-s + 0.328·37-s − 0.312·41-s + 0.609·43-s − 0.894·45-s + 1.16·47-s + 1/7·49-s + 0.824·53-s + 1.07·55-s + 0.768·61-s + 0.377·63-s + 0.496·65-s − 0.488·67-s − 0.949·71-s − 1.17·73-s − 0.455·77-s − 1.80·79-s + 81-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: Trivial
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 \( 1 + p T^{2} \)
5 \( 1 - 2 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 - 8 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 6 T + p T^{2} \)
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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

−14.81121104893668, −14.34868127257753, −13.90206852665696, −13.31839859704330, −12.86855551819810, −12.25374811384644, −11.79963655971808, −11.26168802885191, −10.52642968192778, −10.32285283235831, −9.504751543648239, −9.063185996448153, −8.743837397256738, −8.079438409495132, −7.380052358723173, −6.668174452598185, −6.205923287650602, −5.647922848778993, −5.483305687903253, −4.256982773284892, −3.896352416444046, −3.171443795096741, −2.458930094828518, −1.777710734734662, −1.071113523814379, 0, 1.071113523814379, 1.777710734734662, 2.458930094828518, 3.171443795096741, 3.896352416444046, 4.256982773284892, 5.483305687903253, 5.647922848778993, 6.205923287650602, 6.668174452598185, 7.380052358723173, 8.079438409495132, 8.743837397256738, 9.063185996448153, 9.504751543648239, 10.32285283235831, 10.52642968192778, 11.26168802885191, 11.79963655971808, 12.25374811384644, 12.86855551819810, 13.31839859704330, 13.90206852665696, 14.34868127257753, 14.81121104893668

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