Properties

Degree 1
Conductor $ 7 \cdot 23 $
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  + 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 + 19-s + 20-s − 22-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 + 19-s + 20-s − 22-s − 24-s + 25-s − 26-s − 27-s + 29-s − 30-s − 31-s + 32-s + 33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & \, \Lambda(\chi,1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 161 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & \, \Lambda(1-s,\chi) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(161\)    =    \(7 \cdot 23\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(0\)
character  :  $\chi_{161} (160, \cdot )$
Sato-Tate  :  $\mu(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 161,\ (0:\ ),\ 1)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(1.750625444\)
\(L(\frac12,\chi)\)  \(\approx\)  \(1.750625444\)
\(L(\chi,1)\)  \(\approx\)  \(1.586762750\)
\(L(1,\chi)\)  \(\approx\)  \(1.586762750\)

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

−28.226252687712368459295611300423, −26.77274857872284051945166964378, −25.56632477095717528263813972541, −24.6117817142226311236206426553, −23.79090883058220017624650656748, −22.85859667003473489391548494036, −21.93560577389607203213354983439, −21.3477563724487271676221702084, −20.36655620582330525564999594214, −18.80742558305517790053516488457, −17.717278743183782620940570904444, −16.741384089260543997842076207, −15.902790106330856173014191101230, −14.65074325152942269815807234764, −13.574027095976763595476127239559, −12.647676531253781502250337935563, −11.81400642901997425883153943088, −10.50030792309514014368953394528, −9.87029862675439222505852118235, −7.621130913731959828121210104193, −6.57503960231461631301215816788, −5.36486926249605990713385616200, −4.999082081045909074381129347452, −3.10218030541642144456326707603, −1.65723590715700437225096705131, 1.65723590715700437225096705131, 3.10218030541642144456326707603, 4.999082081045909074381129347452, 5.36486926249605990713385616200, 6.57503960231461631301215816788, 7.621130913731959828121210104193, 9.87029862675439222505852118235, 10.50030792309514014368953394528, 11.81400642901997425883153943088, 12.647676531253781502250337935563, 13.574027095976763595476127239559, 14.65074325152942269815807234764, 15.902790106330856173014191101230, 16.741384089260543997842076207, 17.717278743183782620940570904444, 18.80742558305517790053516488457, 20.36655620582330525564999594214, 21.3477563724487271676221702084, 21.93560577389607203213354983439, 22.85859667003473489391548494036, 23.79090883058220017624650656748, 24.6117817142226311236206426553, 25.56632477095717528263813972541, 26.77274857872284051945166964378, 28.226252687712368459295611300423

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