↑

Properties

Label 2-376712-1.1-c1-0-23
Degree $2$
Conductor $376712$
Sign $-1$
Analytic cond. $3008.06$
Root an. cond. $54.8457$
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 − 3·5-s − 2·9-s + 11-s − 7·13-s − 3·15-s + 5·17-s − 7·19-s + 8·23-s + 4·25-s − 5·27-s + 2·29-s + 33-s − 37-s − 7·39-s − 3·41-s + 7·43-s + 6·45-s + 5·51-s − 9·53-s − 3·55-s − 7·57-s + 5·59-s − 2·61-s + 21·65-s − 7·67-s + 8·69-s + ⋯
L(s)  = 1  + 0.577·3-s − 1.34·5-s − 2/3·9-s + 0.301·11-s − 1.94·13-s − 0.774·15-s + 1.21·17-s − 1.60·19-s + 1.66·23-s + 4/5·25-s − 0.962·27-s + 0.371·29-s + 0.174·33-s − 0.164·37-s − 1.12·39-s − 0.468·41-s + 1.06·43-s + 0.894·45-s + 0.700·51-s − 1.23·53-s − 0.404·55-s − 0.927·57-s + 0.650·59-s − 0.256·61-s + 2.60·65-s − 0.855·67-s + 0.963·69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 376712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 376712 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(376712\)    =    \(2^{3} \cdot 7^{2} \cdot 31^{2}\)
Sign: $-1$
Analytic conductor: \(3008.06\)
Root analytic conductor: \(54.8457\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 376712,\ (\ :1/2),\ -1)\)

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 \( 1 \)
31 \( 1 \)
good3 \( 1 - T + p T^{2} \)
5 \( 1 + 3 T + p T^{2} \)
11 \( 1 - T + p T^{2} \)
13 \( 1 + 7 T + p T^{2} \)
17 \( 1 - 5 T + p T^{2} \)
19 \( 1 + 7 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
37 \( 1 + T + p T^{2} \)
41 \( 1 + 3 T + p T^{2} \)
43 \( 1 - 7 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 9 T + p T^{2} \)
59 \( 1 - 5 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 7 T + p T^{2} \)
71 \( 1 - 5 T + p T^{2} \)
73 \( 1 + 3 T + p T^{2} \)
79 \( 1 + T + p T^{2} \)
83 \( 1 - 3 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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

−12.62141303072717, −12.26578654803187, −11.84894470335995, −11.46286935140753, −10.88911714659664, −10.58875752159030, −9.890897065899626, −9.541177325914472, −8.901553893339274, −8.674517498896181, −8.068683663212103, −7.702389854308835, −7.374653121986739, −6.881399546150609, −6.345753712309180, −5.691099166563419, −5.005101965291085, −4.802342201322469, −4.120356423482903, −3.702340742797722, −3.057483642369515, −2.755813859167669, −2.213029805497598, −1.397998983463609, −0.5568952441700380, 0, 0.5568952441700380, 1.397998983463609, 2.213029805497598, 2.755813859167669, 3.057483642369515, 3.702340742797722, 4.120356423482903, 4.802342201322469, 5.005101965291085, 5.691099166563419, 6.345753712309180, 6.881399546150609, 7.374653121986739, 7.702389854308835, 8.068683663212103, 8.674517498896181, 8.901553893339274, 9.541177325914472, 9.890897065899626, 10.58875752159030, 10.88911714659664, 11.46286935140753, 11.84894470335995, 12.26578654803187, 12.62141303072717

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