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  + 1.85·5-s + 2.41·7-s + 2.49·11-s + 0.116·13-s + 6.12·17-s − 3.34·19-s + 8.19·23-s − 1.57·25-s − 0.0936·29-s + 1.44·31-s + 4.47·35-s − 0.230·37-s + 4.49·41-s + 5.35·43-s − 7.01·47-s − 1.16·49-s − 3.67·53-s + 4.62·55-s + 1.76·59-s + 8.66·61-s + 0.215·65-s − 15.1·67-s + 8.51·71-s + 4.74·73-s + 6.03·77-s − 3.24·79-s + 0.437·83-s + ⋯
L(s)  = 1  + 0.828·5-s + 0.913·7-s + 0.752·11-s + 0.0322·13-s + 1.48·17-s − 0.766·19-s + 1.70·23-s − 0.314·25-s − 0.0173·29-s + 0.260·31-s + 0.756·35-s − 0.0378·37-s + 0.702·41-s + 0.816·43-s − 1.02·47-s − 0.166·49-s − 0.505·53-s + 0.623·55-s + 0.230·59-s + 1.10·61-s + 0.0266·65-s − 1.85·67-s + 1.01·71-s + 0.555·73-s + 0.687·77-s − 0.365·79-s + 0.0480·83-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  =  \((2,\ 6012,\ (\ :1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(3.153796709\)
\(L(\frac12)\)  \(\approx\)  \(3.153796709\)
\(L(\frac{3}{2})\)   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 \( 1 - 1.85T + 5T^{2} \)
7 \( 1 - 2.41T + 7T^{2} \)
11 \( 1 - 2.49T + 11T^{2} \)
13 \( 1 - 0.116T + 13T^{2} \)
17 \( 1 - 6.12T + 17T^{2} \)
19 \( 1 + 3.34T + 19T^{2} \)
23 \( 1 - 8.19T + 23T^{2} \)
29 \( 1 + 0.0936T + 29T^{2} \)
31 \( 1 - 1.44T + 31T^{2} \)
37 \( 1 + 0.230T + 37T^{2} \)
41 \( 1 - 4.49T + 41T^{2} \)
43 \( 1 - 5.35T + 43T^{2} \)
47 \( 1 + 7.01T + 47T^{2} \)
53 \( 1 + 3.67T + 53T^{2} \)
59 \( 1 - 1.76T + 59T^{2} \)
61 \( 1 - 8.66T + 61T^{2} \)
67 \( 1 + 15.1T + 67T^{2} \)
71 \( 1 - 8.51T + 71T^{2} \)
73 \( 1 - 4.74T + 73T^{2} \)
79 \( 1 + 3.24T + 79T^{2} \)
83 \( 1 - 0.437T + 83T^{2} \)
89 \( 1 + 11.9T + 89T^{2} \)
97 \( 1 - 5.72T + 97T^{2} \)
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

−8.083961385446700354558051812897, −7.40987447461993647015141522916, −6.60280662825159296719099004994, −5.90962816279714972419134685941, −5.23897108277763268173849182609, −4.56559163654757471439683260914, −3.63942655137263077135661495446, −2.71028360404887738067007013675, −1.71479057497070911595844381146, −1.03559393555351406408400121008, 1.03559393555351406408400121008, 1.71479057497070911595844381146, 2.71028360404887738067007013675, 3.63942655137263077135661495446, 4.56559163654757471439683260914, 5.23897108277763268173849182609, 5.90962816279714972419134685941, 6.60280662825159296719099004994, 7.40987447461993647015141522916, 8.083961385446700354558051812897

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