Properties

Degree 1
Conductor $ 5 \cdot 643 $
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 + 6-s − 7-s + 8-s + 9-s − 11-s + 12-s + 13-s − 14-s + 16-s + 17-s + 18-s − 19-s − 21-s − 22-s − 23-s + 24-s + 26-s + 27-s − 28-s + 29-s + 31-s + 32-s − 33-s + 34-s + ⋯
L(s,χ)  = 1  + 2-s + 3-s + 4-s + 6-s − 7-s + 8-s + 9-s − 11-s + 12-s + 13-s − 14-s + 16-s + 17-s + 18-s − 19-s − 21-s − 22-s − 23-s + 24-s + 26-s + 27-s − 28-s + 29-s + 31-s + 32-s − 33-s + 34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 3215 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & \, \Lambda(\chi,1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 3215 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & \, \Lambda(1-s,\chi) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(3215\)    =    \(5 \cdot 643\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(0\)
character  :  $\chi_{3215} (3214, \cdot )$
Sato-Tate  :  $\mu(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 3215,\ (1:\ ),\ 1)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(7.449564464\)
\(L(\frac12,\chi)\)  \(\approx\)  \(7.449564464\)
\(L(\chi,1)\)  \(\approx\)  \(2.770316498\)
\(L(1,\chi)\)  \(\approx\)  \(2.770316498\)

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

−18.86342779998923272054752067521, −18.44500246777093104445203221917, −17.148156053956261408824080676653, −16.25314048620783295372729797429, −15.72115292936187578108138688401, −15.387828875802763031542423080420, −14.39796217963025177073580932087, −13.78825372860695615494494091663, −13.30886212644369591831977994986, −12.58818147679978652651534818527, −12.15915748559184923068676347829, −10.90410199316932537945350943244, −10.269382807565392861906795271256, −9.74313534462776086897755016350, −8.56992691124748734616782562969, −7.99127432332524565487274184461, −7.25873643796003191951989758411, −6.28323225154295500981627604008, −5.913919988888241190244185868180, −4.6830298961824691368662257501, −4.03842572117548901557513631479, −3.21013889655314043543029759314, −2.765317950743839927095535983344, −1.89499207178163743146015363360, −0.80834638004106956861801105373, 0.80834638004106956861801105373, 1.89499207178163743146015363360, 2.765317950743839927095535983344, 3.21013889655314043543029759314, 4.03842572117548901557513631479, 4.6830298961824691368662257501, 5.913919988888241190244185868180, 6.28323225154295500981627604008, 7.25873643796003191951989758411, 7.99127432332524565487274184461, 8.56992691124748734616782562969, 9.74313534462776086897755016350, 10.269382807565392861906795271256, 10.90410199316932537945350943244, 12.15915748559184923068676347829, 12.58818147679978652651534818527, 13.30886212644369591831977994986, 13.78825372860695615494494091663, 14.39796217963025177073580932087, 15.387828875802763031542423080420, 15.72115292936187578108138688401, 16.25314048620783295372729797429, 17.148156053956261408824080676653, 18.44500246777093104445203221917, 18.86342779998923272054752067521

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