Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{2} \cdot 167 $
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  + 4.42·5-s + 1.88·7-s − 1.84·11-s − 5.16·13-s − 0.0891·17-s + 6.45·19-s + 3.21·23-s + 14.6·25-s − 3.02·29-s − 10.4·31-s + 8.34·35-s − 1.51·37-s + 7.82·41-s − 7.61·43-s + 11.5·47-s − 3.45·49-s + 2.84·53-s − 8.17·55-s + 5.92·59-s + 4.29·61-s − 22.8·65-s + 13.3·67-s + 11.9·71-s − 0.174·73-s − 3.47·77-s + 2.45·79-s + 11.2·83-s + ⋯
L(s)  = 1  + 1.98·5-s + 0.712·7-s − 0.556·11-s − 1.43·13-s − 0.0216·17-s + 1.48·19-s + 0.669·23-s + 2.92·25-s − 0.560·29-s − 1.87·31-s + 1.41·35-s − 0.248·37-s + 1.22·41-s − 1.16·43-s + 1.68·47-s − 0.492·49-s + 0.390·53-s − 1.10·55-s + 0.770·59-s + 0.550·61-s − 2.83·65-s + 1.62·67-s + 1.41·71-s − 0.0203·73-s − 0.396·77-s + 0.276·79-s + 1.23·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6012 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(6012\)    =    \(2^{2} \cdot 3^{2} \cdot 167\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{6012} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 6012,\ (\ :1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(3.217610040\)
\(L(\frac12)\)  \(\approx\)  \(3.217610040\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;167\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;167\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
167 \( 1 - T \)
good5 \( 1 - 4.42T + 5T^{2} \)
7 \( 1 - 1.88T + 7T^{2} \)
11 \( 1 + 1.84T + 11T^{2} \)
13 \( 1 + 5.16T + 13T^{2} \)
17 \( 1 + 0.0891T + 17T^{2} \)
19 \( 1 - 6.45T + 19T^{2} \)
23 \( 1 - 3.21T + 23T^{2} \)
29 \( 1 + 3.02T + 29T^{2} \)
31 \( 1 + 10.4T + 31T^{2} \)
37 \( 1 + 1.51T + 37T^{2} \)
41 \( 1 - 7.82T + 41T^{2} \)
43 \( 1 + 7.61T + 43T^{2} \)
47 \( 1 - 11.5T + 47T^{2} \)
53 \( 1 - 2.84T + 53T^{2} \)
59 \( 1 - 5.92T + 59T^{2} \)
61 \( 1 - 4.29T + 61T^{2} \)
67 \( 1 - 13.3T + 67T^{2} \)
71 \( 1 - 11.9T + 71T^{2} \)
73 \( 1 + 0.174T + 73T^{2} \)
79 \( 1 - 2.45T + 79T^{2} \)
83 \( 1 - 11.2T + 83T^{2} \)
89 \( 1 - 18.3T + 89T^{2} \)
97 \( 1 + 3.13T + 97T^{2} \)
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

−7.976641428206304868998559703545, −7.30146685908226046610378113753, −6.73296641852211256184920743628, −5.63874689075100056905657896878, −5.27496677003415711122665753874, −4.90490666378709603930894278170, −3.49050390440047606152257497226, −2.42492235863041341943804309631, −2.06116010626006796329874322884, −0.973076977190740593462311448105, 0.973076977190740593462311448105, 2.06116010626006796329874322884, 2.42492235863041341943804309631, 3.49050390440047606152257497226, 4.90490666378709603930894278170, 5.27496677003415711122665753874, 5.63874689075100056905657896878, 6.73296641852211256184920743628, 7.30146685908226046610378113753, 7.976641428206304868998559703545

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