Properties

Label 2-47096-1.1-c1-0-6
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

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 5-s − 7-s − 2·9-s − 6·11-s − 3·13-s − 15-s + 2·17-s + 7·19-s − 21-s + 3·23-s − 4·25-s − 5·27-s + 4·31-s − 6·33-s + 35-s + 4·37-s − 3·39-s − 12·41-s + 2·45-s + 6·47-s + 49-s + 2·51-s − 12·53-s + 6·55-s + 7·57-s + 5·59-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.447·5-s − 0.377·7-s − 2/3·9-s − 1.80·11-s − 0.832·13-s − 0.258·15-s + 0.485·17-s + 1.60·19-s − 0.218·21-s + 0.625·23-s − 4/5·25-s − 0.962·27-s + 0.718·31-s − 1.04·33-s + 0.169·35-s + 0.657·37-s − 0.480·39-s − 1.87·41-s + 0.298·45-s + 0.875·47-s + 1/7·49-s + 0.280·51-s − 1.64·53-s + 0.809·55-s + 0.927·57-s + 0.650·59-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 \( 1 - T + p T^{2} \)
5 \( 1 + T + p T^{2} \)
11 \( 1 + 6 T + p T^{2} \)
13 \( 1 + 3 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 7 T + p T^{2} \)
23 \( 1 - 3 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 + 12 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 + 12 T + p T^{2} \)
59 \( 1 - 5 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 10 T + p T^{2} \)
71 \( 1 - 13 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - 17 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 16 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−14.94370970343000, −14.23749528944227, −13.79420746721551, −13.44573186372413, −12.84261477117489, −12.24834044692165, −11.82955478319465, −11.26032532069227, −10.69422848021728, −10.02209005089119, −9.661727088284015, −9.189480902652667, −8.301802594490490, −7.982316032205883, −7.608915410621466, −7.067307250706216, −6.250618183339016, −5.467187525522963, −5.192101495198611, −4.580450507164488, −3.478263295971162, −3.220468666919959, −2.628483081322078, −2.025758492566862, −0.7870963226565282, 0, 0.7870963226565282, 2.025758492566862, 2.628483081322078, 3.220468666919959, 3.478263295971162, 4.580450507164488, 5.192101495198611, 5.467187525522963, 6.250618183339016, 7.067307250706216, 7.608915410621466, 7.982316032205883, 8.301802594490490, 9.189480902652667, 9.661727088284015, 10.02209005089119, 10.69422848021728, 11.26032532069227, 11.82955478319465, 12.24834044692165, 12.84261477117489, 13.44573186372413, 13.79420746721551, 14.23749528944227, 14.94370970343000

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