Properties

Label 2-47432-1.1-c1-0-14
Degree $2$
Conductor $47432$
Sign $-1$
Analytic cond. $378.746$
Root an. cond. $19.4614$
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 − 3·9-s + 2·13-s − 6·17-s + 8·19-s − 25-s − 6·29-s − 8·31-s − 2·37-s + 2·41-s + 4·43-s + 6·45-s + 8·47-s + 6·53-s − 6·61-s − 4·65-s − 4·67-s − 8·71-s + 10·73-s − 16·79-s + 9·81-s + 8·83-s + 12·85-s + 6·89-s − 16·95-s + 6·97-s + 101-s + ⋯
L(s)  = 1  − 0.894·5-s − 9-s + 0.554·13-s − 1.45·17-s + 1.83·19-s − 1/5·25-s − 1.11·29-s − 1.43·31-s − 0.328·37-s + 0.312·41-s + 0.609·43-s + 0.894·45-s + 1.16·47-s + 0.824·53-s − 0.768·61-s − 0.496·65-s − 0.488·67-s − 0.949·71-s + 1.17·73-s − 1.80·79-s + 81-s + 0.878·83-s + 1.30·85-s + 0.635·89-s − 1.64·95-s + 0.609·97-s + 0.0995·101-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 47432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 47432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(47432\)    =    \(2^{3} \cdot 7^{2} \cdot 11^{2}\)
Sign: $-1$
Analytic conductor: \(378.746\)
Root analytic conductor: \(19.4614\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{47432} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 47432,\ (\ :1/2),\ -1)\)

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 \( 1 \)
11 \( 1 \)
good3 \( 1 + p T^{2} \)
5 \( 1 + 2 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
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

−14.97822137195201, −14.23188892477808, −13.86819752241102, −13.35656465880697, −12.80284113227273, −12.14141484508642, −11.62663857721105, −11.27462377850560, −10.94933970364976, −10.26465472616726, −9.442705507145547, −8.890252633008921, −8.775505252044841, −7.856929288159684, −7.408882342237967, −7.111867070753838, −6.049099837683110, −5.803221265557578, −5.094594572345357, −4.418753248483483, −3.661525480578208, −3.404035307126002, −2.527365855570781, −1.832762336945072, −0.7744045392985993, 0, 0.7744045392985993, 1.832762336945072, 2.527365855570781, 3.404035307126002, 3.661525480578208, 4.418753248483483, 5.094594572345357, 5.803221265557578, 6.049099837683110, 7.111867070753838, 7.408882342237967, 7.856929288159684, 8.775505252044841, 8.890252633008921, 9.442705507145547, 10.26465472616726, 10.94933970364976, 11.27462377850560, 11.62663857721105, 12.14141484508642, 12.80284113227273, 13.35656465880697, 13.86819752241102, 14.23188892477808, 14.97822137195201

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