Properties

Degree 1
Conductor 131
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 + 7-s − 8-s + 9-s − 10-s + 11-s + 12-s + 13-s − 14-s + 15-s + 16-s − 17-s − 18-s − 19-s + 20-s + 21-s − 22-s − 23-s − 24-s + 25-s − 26-s + 27-s + 28-s + ⋯
L(s,χ)  = 1  − 2-s + 3-s + 4-s + 5-s − 6-s + 7-s − 8-s + 9-s − 10-s + 11-s + 12-s + 13-s − 14-s + 15-s + 16-s − 17-s − 18-s − 19-s + 20-s + 21-s − 22-s − 23-s − 24-s + 25-s − 26-s + 27-s + 28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 131 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & \, \Lambda(\chi,1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 131 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & \, \Lambda(1-s,\chi) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(131\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(0\)
character  :  $\chi_{131} (130, \cdot )$
Sato-Tate  :  $\mu(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(1,\ 131,\ (1:\ ),\ 1)$
$L(\chi,\frac{1}{2})$  $\approx$  $2.291791789$
$L(\frac12,\chi)$  $\approx$  $2.291791789$
$L(\chi,1)$  $\approx$  1.372411122
$L(1,\chi)$  $\approx$  1.372411122

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

−28.10053949405797751497597689666, −27.48206819897930603478162789249, −26.245709990949587501415297353323, −25.6690023817840441384554054315, −24.68901705426192396383996429606, −24.11122655930628007262105883821, −21.95900116053961313259307203986, −21.00104166473254986579642222624, −20.380788848358368485912569010171, −19.28727450011525264236117569905, −18.15735922290362779426573176589, −17.52780699903546098655249814911, −16.263479424985839767131418457035, −14.95146648574272700902756715717, −14.1946818883451423750950666860, −12.94207615206425877784825218353, −11.324648694991233915382397898150, −10.30229271069095515513599367810, −9.01938987123242348384332874155, −8.597781733817818130966263691234, −7.18696907872269337754588076888, −6.00365414625244325266107823244, −3.99039985431927840912030524596, −2.19320470018265180013840669684, −1.48998515302936687800012193825, 1.48998515302936687800012193825, 2.19320470018265180013840669684, 3.99039985431927840912030524596, 6.00365414625244325266107823244, 7.18696907872269337754588076888, 8.597781733817818130966263691234, 9.01938987123242348384332874155, 10.30229271069095515513599367810, 11.324648694991233915382397898150, 12.94207615206425877784825218353, 14.1946818883451423750950666860, 14.95146648574272700902756715717, 16.263479424985839767131418457035, 17.52780699903546098655249814911, 18.15735922290362779426573176589, 19.28727450011525264236117569905, 20.380788848358368485912569010171, 21.00104166473254986579642222624, 21.95900116053961313259307203986, 24.11122655930628007262105883821, 24.68901705426192396383996429606, 25.6690023817840441384554054315, 26.245709990949587501415297353323, 27.48206819897930603478162789249, 28.10053949405797751497597689666

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