Properties

Degree 2
Conductor $ 3^{2} \cdot 5 \cdot 7^{2} $
Sign $1$
Motivic weight 3
Primitive yes
Self-dual yes
Analytic rank 0

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 0.236·2-s − 7.94·4-s − 5·5-s + 3.76·8-s + 1.18·10-s + 50.4·11-s + 80.9·13-s + 62.6·16-s + 76.3·17-s − 4.13·19-s + 39.7·20-s − 11.9·22-s + 204.·23-s + 25·25-s − 19.1·26-s + 91.1·29-s − 198.·31-s − 44.9·32-s − 18.0·34-s + 155.·37-s + 0.976·38-s − 18.8·40-s − 156.·41-s + 354.·43-s − 400.·44-s − 48.3·46-s − 175.·47-s + ⋯
L(s)  = 1  − 0.0834·2-s − 0.993·4-s − 0.447·5-s + 0.166·8-s + 0.0373·10-s + 1.38·11-s + 1.72·13-s + 0.979·16-s + 1.08·17-s − 0.0499·19-s + 0.444·20-s − 0.115·22-s + 1.85·23-s + 0.200·25-s − 0.144·26-s + 0.583·29-s − 1.14·31-s − 0.248·32-s − 0.0909·34-s + 0.691·37-s + 0.00416·38-s − 0.0743·40-s − 0.597·41-s + 1.25·43-s − 1.37·44-s − 0.154·46-s − 0.545·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(2205\)    =    \(3^{2} \cdot 5 \cdot 7^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(3\)
character  :  $\chi_{2205} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 2205,\ (\ :3/2),\ 1)\)
\(L(2)\)  \(\approx\)  \(2.183956330\)
\(L(\frac12)\)  \(\approx\)  \(2.183956330\)
\(L(\frac{5}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;5,\;7\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{3,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 + 5T \)
7 \( 1 \)
good2 \( 1 + 0.236T + 8T^{2} \)
11 \( 1 - 50.4T + 1.33e3T^{2} \)
13 \( 1 - 80.9T + 2.19e3T^{2} \)
17 \( 1 - 76.3T + 4.91e3T^{2} \)
19 \( 1 + 4.13T + 6.85e3T^{2} \)
23 \( 1 - 204.T + 1.21e4T^{2} \)
29 \( 1 - 91.1T + 2.43e4T^{2} \)
31 \( 1 + 198.T + 2.97e4T^{2} \)
37 \( 1 - 155.T + 5.06e4T^{2} \)
41 \( 1 + 156.T + 6.89e4T^{2} \)
43 \( 1 - 354.T + 7.95e4T^{2} \)
47 \( 1 + 175.T + 1.03e5T^{2} \)
53 \( 1 + 200.T + 1.48e5T^{2} \)
59 \( 1 + 312.T + 2.05e5T^{2} \)
61 \( 1 - 154.T + 2.26e5T^{2} \)
67 \( 1 - 734.T + 3.00e5T^{2} \)
71 \( 1 - 678.T + 3.57e5T^{2} \)
73 \( 1 - 60.8T + 3.89e5T^{2} \)
79 \( 1 + 1.28e3T + 4.93e5T^{2} \)
83 \( 1 - 116.T + 5.71e5T^{2} \)
89 \( 1 + 916.T + 7.04e5T^{2} \)
97 \( 1 - 1.41e3T + 9.12e5T^{2} \)
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

−8.802448582514816043469103332116, −8.126125293156974374046216151301, −7.20817147921881462528039021615, −6.31991108139025137867890261657, −5.51564139556199505158790391789, −4.56847575804599415743294701134, −3.73896023489246594250907191974, −3.26535449115075605289438094617, −1.34550982591600312362860355750, −0.817091583680607686358157390539, 0.817091583680607686358157390539, 1.34550982591600312362860355750, 3.26535449115075605289438094617, 3.73896023489246594250907191974, 4.56847575804599415743294701134, 5.51564139556199505158790391789, 6.31991108139025137867890261657, 7.20817147921881462528039021615, 8.126125293156974374046216151301, 8.802448582514816043469103332116

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