Properties

Label 2-3920-1.1-c1-0-36
Degree $2$
Conductor $3920$
Sign $1$
Analytic cond. $31.3013$
Root an. cond. $5.59476$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 5-s − 2·9-s + 6·11-s + 4·13-s + 15-s + 2·19-s + 3·23-s + 25-s − 5·27-s − 3·29-s + 8·31-s + 6·33-s − 4·37-s + 4·39-s − 9·41-s + 7·43-s − 2·45-s − 6·53-s + 6·55-s + 2·57-s − 6·59-s − 5·61-s + 4·65-s − 5·67-s + 3·69-s + 6·71-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.447·5-s − 2/3·9-s + 1.80·11-s + 1.10·13-s + 0.258·15-s + 0.458·19-s + 0.625·23-s + 1/5·25-s − 0.962·27-s − 0.557·29-s + 1.43·31-s + 1.04·33-s − 0.657·37-s + 0.640·39-s − 1.40·41-s + 1.06·43-s − 0.298·45-s − 0.824·53-s + 0.809·55-s + 0.264·57-s − 0.781·59-s − 0.640·61-s + 0.496·65-s − 0.610·67-s + 0.361·69-s + 0.712·71-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3920\)    =    \(2^{4} \cdot 5 \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(31.3013\)
Root analytic conductor: \(5.59476\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{3920} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 3920,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.981430920\)
\(L(\frac12)\) \(\approx\) \(2.981430920\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 - T \)
7 \( 1 \)
good3 \( 1 - T + p T^{2} \)
11 \( 1 - 6 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 - 3 T + p T^{2} \)
29 \( 1 + 3 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 + 9 T + p T^{2} \)
43 \( 1 - 7 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 + 5 T + p T^{2} \)
67 \( 1 + 5 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 - 16 T + p T^{2} \)
79 \( 1 + 2 T + p T^{2} \)
83 \( 1 - 3 T + p T^{2} \)
89 \( 1 - 15 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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

−8.585611469505637779917975201697, −7.922123577593787592042594371067, −6.84529302039872832385406815476, −6.32452963481278771786211763193, −5.63095877070288112514665916724, −4.60530874348065801098420211527, −3.63703389202489724923396682704, −3.13138504872146264156164214775, −1.92318340352558472695927844813, −1.04621856322269760220444445166, 1.04621856322269760220444445166, 1.92318340352558472695927844813, 3.13138504872146264156164214775, 3.63703389202489724923396682704, 4.60530874348065801098420211527, 5.63095877070288112514665916724, 6.32452963481278771786211763193, 6.84529302039872832385406815476, 7.922123577593787592042594371067, 8.585611469505637779917975201697

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