Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{2} \cdot 7 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + 7-s + 6·11-s + 2·13-s − 4·19-s + 6·23-s − 5·25-s − 6·29-s + 8·31-s + 2·37-s − 12·41-s − 4·43-s − 12·47-s + 49-s + 6·53-s − 10·61-s + 8·67-s − 6·71-s − 10·73-s + 6·77-s − 4·79-s + 12·83-s − 12·89-s + 2·91-s − 10·97-s + 12·101-s + 8·103-s + 6·107-s + ⋯
L(s)  = 1  + 0.377·7-s + 1.80·11-s + 0.554·13-s − 0.917·19-s + 1.25·23-s − 25-s − 1.11·29-s + 1.43·31-s + 0.328·37-s − 1.87·41-s − 0.609·43-s − 1.75·47-s + 1/7·49-s + 0.824·53-s − 1.28·61-s + 0.977·67-s − 0.712·71-s − 1.17·73-s + 0.683·77-s − 0.450·79-s + 1.31·83-s − 1.27·89-s + 0.209·91-s − 1.01·97-s + 1.19·101-s + 0.788·103-s + 0.580·107-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{252} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 252,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.37979$
$L(\frac12)$  $\approx$  $1.37979$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;7\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 - T \)
good5 \( 1 + p T^{2} \)
11 \( 1 - 6 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 12 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 12 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
show more
show less
\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−11.78681575790007246498903350804, −11.34449628375814460891723631889, −10.10483686416197845980665091731, −9.072046417107109896038765861128, −8.300043091642042619777544931378, −6.93446262615281462668117301613, −6.11056271857714724112691707067, −4.63917458628061830491997962672, −3.53462971911322619523210724383, −1.58867103529813382887688476545, 1.58867103529813382887688476545, 3.53462971911322619523210724383, 4.63917458628061830491997962672, 6.11056271857714724112691707067, 6.93446262615281462668117301613, 8.300043091642042619777544931378, 9.072046417107109896038765861128, 10.10483686416197845980665091731, 11.34449628375814460891723631889, 11.78681575790007246498903350804

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