Properties

Degree 1
Conductor $ 7 \cdot 19 $
Sign $1$
Motivic weight 0
Primitive yes
Self-dual yes
Analytic rank 0

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Normalization:  

Dirichlet series

L(χ,s)  = 1  − 2-s + 3-s + 4-s − 5-s − 6-s − 8-s + 9-s + 10-s + 11-s + 12-s + 13-s − 15-s + 16-s − 17-s − 18-s − 20-s − 22-s + 23-s − 24-s + 25-s − 26-s + 27-s − 29-s + 30-s + 31-s − 32-s + 33-s + ⋯
L(s,χ)  = 1  − 2-s + 3-s + 4-s − 5-s − 6-s − 8-s + 9-s + 10-s + 11-s + 12-s + 13-s − 15-s + 16-s − 17-s − 18-s − 20-s − 22-s + 23-s − 24-s + 25-s − 26-s + 27-s − 29-s + 30-s + 31-s − 32-s + 33-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 133 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & \, \Lambda(\chi,1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 133 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & \, \Lambda(1-s,\chi) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(133\)    =    \(7 \cdot 19\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(0\)
character  :  $\chi_{133} (132, \cdot )$
Sato-Tate  :  $\mu(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(1,\ 133,\ (0:\ ),\ 1)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.9084590317$
$L(\frac12,\chi)$  $\approx$  $0.9084590317$
$L(\chi,1)$  $\approx$  0.8936883087
$L(1,\chi)$  $\approx$  0.8936883087

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

−28.24758837664794843697727615596, −27.46824532055970711680240660253, −26.66137132762555261058646946522, −25.873482625399781992545440174899, −24.77625639764276167769021135215, −24.11795044361899509082713214847, −22.65193510260904049207325830085, −21.08112790359596772335582481385, −20.28573769920505928756350109642, −19.39531025590113440579401678424, −18.846640952188759442978194100199, −17.56045729012685395588945963231, −16.19422064040168239364255689780, −15.45340298751968508614649748391, −14.52780433872581024357404687792, −13.00238459337027161207626894526, −11.671390658087327129590431380805, −10.72220816155185961236418829983, −9.18601306679871263617429376743, −8.622107034578946389255971755361, −7.50667577710357321372913226853, −6.537290780686861531090829493660, −4.151323040637174249382431018531, −3.02676951049389218311006396684, −1.39047300131328585995249715062, 1.39047300131328585995249715062, 3.02676951049389218311006396684, 4.151323040637174249382431018531, 6.537290780686861531090829493660, 7.50667577710357321372913226853, 8.622107034578946389255971755361, 9.18601306679871263617429376743, 10.72220816155185961236418829983, 11.671390658087327129590431380805, 13.00238459337027161207626894526, 14.52780433872581024357404687792, 15.45340298751968508614649748391, 16.19422064040168239364255689780, 17.56045729012685395588945963231, 18.846640952188759442978194100199, 19.39531025590113440579401678424, 20.28573769920505928756350109642, 21.08112790359596772335582481385, 22.65193510260904049207325830085, 24.11795044361899509082713214847, 24.77625639764276167769021135215, 25.873482625399781992545440174899, 26.66137132762555261058646946522, 27.46824532055970711680240660253, 28.24758837664794843697727615596

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