Properties

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

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  − 3-s + 7-s + 9-s + 11-s + 13-s − 17-s + 19-s − 21-s + 23-s − 27-s − 29-s − 31-s − 33-s + 37-s − 39-s + 41-s − 43-s + 47-s + 49-s + 51-s + 53-s − 57-s + 59-s − 61-s + 63-s − 67-s − 69-s + ⋯
L(s,χ)  = 1  − 3-s + 7-s + 9-s + 11-s + 13-s − 17-s + 19-s − 21-s + 23-s − 27-s − 29-s − 31-s − 33-s + 37-s − 39-s + 41-s − 43-s + 47-s + 49-s + 51-s + 53-s − 57-s + 59-s − 61-s + 63-s − 67-s − 69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 40 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & \, \Lambda(\chi,1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 40 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & \, \Lambda(1-s,\chi) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(40\)    =    \(2^{3} \cdot 5\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(0\)
character  :  $\chi_{40} (19, \cdot )$
Sato-Tate  :  $\mu(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 40,\ (1:\ ),\ 1)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(1.273701528\)
\(L(\frac12,\chi)\)  \(\approx\)  \(1.273701528\)
\(L(\chi,1)\)  \(\approx\)  \(0.9934588265\)
\(L(1,\chi)\)  \(\approx\)  \(0.9934588265\)

Euler product

\[\begin{aligned}L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1}\end{aligned}\]
\[\begin{aligned}L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−34.89620878693812615394724706126, −33.47177184188358054648343802471, −32.95373110336489202294647741903, −31.000878238848054813734736914031, −30.08283072310760662654485254361, −28.75620407320413245889633013070, −27.74001549128314964333399015308, −26.79999431487267833684377135315, −24.91164474281862795156104112947, −23.94125759717354704293742119293, −22.71111814469318191049357358124, −21.62820070972895871809325135529, −20.30870606713841120462503925791, −18.50131456917208069943603260787, −17.56198392586458758656427315753, −16.38749542986661449060865447244, −14.9334623468118711546352731953, −13.31089863478015673369236422861, −11.66929573512790455200503900855, −10.93602082108279406030214194985, −9.06078728593525702719913749440, −7.19780040148562061479699240603, −5.69697314536849519104203193886, −4.211079232419059888877011668122, −1.32058972400961870327719439444, 1.32058972400961870327719439444, 4.211079232419059888877011668122, 5.69697314536849519104203193886, 7.19780040148562061479699240603, 9.06078728593525702719913749440, 10.93602082108279406030214194985, 11.66929573512790455200503900855, 13.31089863478015673369236422861, 14.9334623468118711546352731953, 16.38749542986661449060865447244, 17.56198392586458758656427315753, 18.50131456917208069943603260787, 20.30870606713841120462503925791, 21.62820070972895871809325135529, 22.71111814469318191049357358124, 23.94125759717354704293742119293, 24.91164474281862795156104112947, 26.79999431487267833684377135315, 27.74001549128314964333399015308, 28.75620407320413245889633013070, 30.08283072310760662654485254361, 31.000878238848054813734736914031, 32.95373110336489202294647741903, 33.47177184188358054648343802471, 34.89620878693812615394724706126

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