↑

Properties

Label 2-2940-1.1-c1-0-25
Degree $2$
Conductor $2940$
Sign $-1$
Analytic cond. $23.4760$
Root an. cond. $4.84520$
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 + 9-s − 6·11-s + 15-s − 6·17-s − 4·19-s + 6·23-s + 25-s + 27-s − 2·29-s + 8·31-s − 6·33-s − 2·37-s − 10·41-s − 12·43-s + 45-s − 8·47-s − 6·51-s − 2·53-s − 6·55-s − 4·57-s − 4·59-s + 8·61-s − 16·67-s + 6·69-s − 10·71-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.447·5-s + 1/3·9-s − 1.80·11-s + 0.258·15-s − 1.45·17-s − 0.917·19-s + 1.25·23-s + 1/5·25-s + 0.192·27-s − 0.371·29-s + 1.43·31-s − 1.04·33-s − 0.328·37-s − 1.56·41-s − 1.82·43-s + 0.149·45-s − 1.16·47-s − 0.840·51-s − 0.274·53-s − 0.809·55-s − 0.529·57-s − 0.520·59-s + 1.02·61-s − 1.95·67-s + 0.722·69-s − 1.18·71-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2940 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2940 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2940\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 7^{2}\)
Sign: $-1$
Analytic conductor: \(23.4760\)
Root analytic conductor: \(4.84520\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 2940,\ (\ :1/2),\ -1)\)

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 \( 1 \)
3 \( 1 - T \)
5 \( 1 - T \)
7 \( 1 \)
good11 \( 1 + 6 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 + 12 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 + 16 T + p T^{2} \)
71 \( 1 + 10 T + p T^{2} \)
73 \( 1 + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 8 T + p T^{2} \)
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

−8.525303111443645771284234505754, −7.72943844024026480595746526403, −6.85246807995857474261072430348, −6.23348028516058929219254867338, −4.99179962910816407781567454718, −4.72448120718678638109389073670, −3.32486528276801553350819511211, −2.59228944046209387754945520849, −1.78387510488487297674238584819, 0, 1.78387510488487297674238584819, 2.59228944046209387754945520849, 3.32486528276801553350819511211, 4.72448120718678638109389073670, 4.99179962910816407781567454718, 6.23348028516058929219254867338, 6.85246807995857474261072430348, 7.72943844024026480595746526403, 8.525303111443645771284234505754

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