Properties

Degree 1
Conductor $ 3 \cdot 37 $
Sign $1$
Motivic weight 0
Primitive yes
Self-dual yes
Analytic rank 0

Related objects

Learn more about

Normalization:  

Dirichlet series

L(χ,s)  = 1  + 2-s + 4-s + 5-s + 7-s + 8-s + 10-s − 11-s − 13-s + 14-s + 16-s + 17-s − 19-s + 20-s − 22-s + 23-s + 25-s − 26-s + 28-s + 29-s − 31-s + 32-s + 34-s + 35-s − 38-s + 40-s − 41-s − 43-s + ⋯
L(s,χ)  = 1  + 2-s + 4-s + 5-s + 7-s + 8-s + 10-s − 11-s − 13-s + 14-s + 16-s + 17-s − 19-s + 20-s − 22-s + 23-s + 25-s − 26-s + 28-s + 29-s − 31-s + 32-s + 34-s + 35-s − 38-s + 40-s − 41-s − 43-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 111 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & \, \Lambda(\chi,1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 111 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & \, \Lambda(1-s,\chi) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(111\)    =    \(3 \cdot 37\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(0\)
character  :  $\chi_{111} (110, \cdot )$
Sato-Tate  :  $\mu(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(1,\ 111,\ (1:\ ),\ 1)$
$L(\chi,\frac{1}{2})$  $\approx$  $4.018487349$
$L(\frac12,\chi)$  $\approx$  $4.018487349$
$L(\chi,1)$  $\approx$  2.385494229
$L(1,\chi)$  $\approx$  2.385494229

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

−29.410783343940605735060334563732, −28.55044247075892998197504751996, −27.153648614558493521094926481049, −25.75995499611335003359892012389, −24.993543275966750753966435287570, −23.99151932841177520078080327008, −23.13634028367934168997612094373, −21.67013488419763618327894787851, −21.281649104546924705742566477586, −20.32450705058597914708587668926, −18.824051675795694853728087854558, −17.49579586395880586962990542039, −16.602485150507739228363434008960, −15.03710621653526779194013713608, −14.392261552192090746786738833462, −13.26982252916463142779127787459, −12.32527336261593817132751592862, −10.94954657241707780180907292524, −10.02466677956892528031856770206, −8.17757446396435144676603620098, −6.90522776954867651394444893962, −5.44443005958836874755376360026, −4.80339932893683508528076265259, −2.86468766368800305685819078989, −1.71181386932933385219773230284, 1.71181386932933385219773230284, 2.86468766368800305685819078989, 4.80339932893683508528076265259, 5.44443005958836874755376360026, 6.90522776954867651394444893962, 8.17757446396435144676603620098, 10.02466677956892528031856770206, 10.94954657241707780180907292524, 12.32527336261593817132751592862, 13.26982252916463142779127787459, 14.392261552192090746786738833462, 15.03710621653526779194013713608, 16.602485150507739228363434008960, 17.49579586395880586962990542039, 18.824051675795694853728087854558, 20.32450705058597914708587668926, 21.281649104546924705742566477586, 21.67013488419763618327894787851, 23.13634028367934168997612094373, 23.99151932841177520078080327008, 24.993543275966750753966435287570, 25.75995499611335003359892012389, 27.153648614558493521094926481049, 28.55044247075892998197504751996, 29.410783343940605735060334563732

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