Properties

Label 2-47096-1.1-c1-0-2
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 7-s − 2·9-s + 6·11-s + 4·13-s − 6·17-s + 5·19-s + 21-s − 23-s − 5·25-s + 5·27-s − 8·31-s − 6·33-s + 8·37-s − 4·39-s + 7·41-s + 10·43-s + 5·47-s + 49-s + 6·51-s − 11·53-s − 5·57-s − 2·61-s + 2·63-s + 9·67-s + 69-s − 15·71-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.377·7-s − 2/3·9-s + 1.80·11-s + 1.10·13-s − 1.45·17-s + 1.14·19-s + 0.218·21-s − 0.208·23-s − 25-s + 0.962·27-s − 1.43·31-s − 1.04·33-s + 1.31·37-s − 0.640·39-s + 1.09·41-s + 1.52·43-s + 0.729·47-s + 1/7·49-s + 0.840·51-s − 1.51·53-s − 0.662·57-s − 0.256·61-s + 0.251·63-s + 1.09·67-s + 0.120·69-s − 1.78·71-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: \(0\)
Selberg data: \((2,\ 47096,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.798713992\)
\(L(\frac12)\) \(\approx\) \(1.798713992\)
\(L(\frac{3}{2})\) 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 \( 1 + T + p T^{2} \)
5 \( 1 + p T^{2} \)
11 \( 1 - 6 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 5 T + p T^{2} \)
23 \( 1 + T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 - 7 T + p T^{2} \)
43 \( 1 - 10 T + p T^{2} \)
47 \( 1 - 5 T + p T^{2} \)
53 \( 1 + 11 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 9 T + p T^{2} \)
71 \( 1 + 15 T + p T^{2} \)
73 \( 1 + 3 T + p T^{2} \)
79 \( 1 - 12 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 + 11 T + p T^{2} \)
97 \( 1 - T + p T^{2} \)
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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

−14.49072662590222, −14.07878733017627, −13.62272920134006, −13.08206594738746, −12.39066801401223, −11.99581575586101, −11.29594073734168, −11.15800367633197, −10.75436363315903, −9.665840393103859, −9.329601043030239, −8.993953366962578, −8.378389674443431, −7.599511795469580, −7.052527309999259, −6.357546604297677, −5.985491353560417, −5.716248059710127, −4.694022003355096, −4.080388013744192, −3.685966503025467, −2.907116018830050, −2.060795406203011, −1.246832398091718, −0.5410907872285886, 0.5410907872285886, 1.246832398091718, 2.060795406203011, 2.907116018830050, 3.685966503025467, 4.080388013744192, 4.694022003355096, 5.716248059710127, 5.985491353560417, 6.357546604297677, 7.052527309999259, 7.599511795469580, 8.378389674443431, 8.993953366962578, 9.329601043030239, 9.665840393103859, 10.75436363315903, 11.15800367633197, 11.29594073734168, 11.99581575586101, 12.39066801401223, 13.08206594738746, 13.62272920134006, 14.07878733017627, 14.49072662590222

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