Properties

Degree 1
Conductor $ 7 \cdot 11 $
Sign $0.303 - 0.952i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.669 + 0.743i)2-s + (0.913 − 0.406i)3-s + (−0.104 − 0.994i)4-s + (−0.978 + 0.207i)5-s + (−0.309 + 0.951i)6-s + (0.809 + 0.587i)8-s + (0.669 − 0.743i)9-s + (0.5 − 0.866i)10-s + (−0.5 − 0.866i)12-s + (−0.309 − 0.951i)13-s + (−0.809 + 0.587i)15-s + (−0.978 + 0.207i)16-s + (−0.669 − 0.743i)17-s + (0.104 + 0.994i)18-s + (0.104 − 0.994i)19-s + (0.309 + 0.951i)20-s + ⋯
L(s,χ)  = 1  + (−0.669 + 0.743i)2-s + (0.913 − 0.406i)3-s + (−0.104 − 0.994i)4-s + (−0.978 + 0.207i)5-s + (−0.309 + 0.951i)6-s + (0.809 + 0.587i)8-s + (0.669 − 0.743i)9-s + (0.5 − 0.866i)10-s + (−0.5 − 0.866i)12-s + (−0.309 − 0.951i)13-s + (−0.809 + 0.587i)15-s + (−0.978 + 0.207i)16-s + (−0.669 − 0.743i)17-s + (0.104 + 0.994i)18-s + (0.104 − 0.994i)19-s + (0.309 + 0.951i)20-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (0.303 - 0.952i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 77 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (0.303 - 0.952i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(77\)    =    \(7 \cdot 11\)
\( \varepsilon \)  =  $0.303 - 0.952i$
motivic weight  =  \(0\)
character  :  $\chi_{77} (51, \cdot )$
Sato-Tate  :  $\mu(30)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 77,\ (1:\ ),\ 0.303 - 0.952i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.7928479756 - 0.5797761440i$
$L(\frac12,\chi)$  $\approx$  $0.7928479756 - 0.5797761440i$
$L(\chi,1)$  $\approx$  0.8211013048 - 0.06594312912i
$L(1,\chi)$  $\approx$  0.8211013048 - 0.06594312912i

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

−31.2448917499912842329795282608, −30.315001712051873248789964427664, −28.94432995464327089082348404978, −27.79122571942636409026580420289, −26.98143663390834772853279921599, −26.23708518706473594277534784411, −25.07144864402516504758648468351, −23.7011875653759174960256279003, −22.07336619399538721633362815333, −21.16507035102140867614207196232, −19.96622634016099908100353435515, −19.47865125851700053316589241593, −18.37832294698605624840881493047, −16.6934108884377241721220161038, −15.80537915925150585660308154646, −14.416254036510179789040760224997, −12.96645474762810963162742044019, −11.783481284240019302951210843888, −10.54187310197458927121001253411, −9.26169698053530304472236409980, −8.31765251726373417617990605451, −7.28453675039893634592049072428, −4.38647334471547579119703519842, −3.47331796367863675772417375771, −1.81695266124800131173800294883, 0.54138558364351828696127675316, 2.69356026270233266803972238867, 4.56743437280845701743562319239, 6.608497143955168168463948665144, 7.64172586394826919537503488031, 8.4775289584696221283912251184, 9.74331546473643195374321013183, 11.246553968154707826081344973261, 12.91587323596839301210636094905, 14.295059613411615199208644851439, 15.24628303126628488663331775743, 16.03754718943321908166078067258, 17.7447400050455931622396812882, 18.62337278682156826704767915410, 19.75453245360620663351609732122, 20.26790112006270681117623106372, 22.39147159161146804328333278243, 23.62332735750600771368025761449, 24.476519644916477616108258796350, 25.41034994462262817810722207993, 26.58933955153761969488378424050, 27.11496730933831861469220514257, 28.381529513767545187615336186674, 29.82245419479349070562553025702, 30.9358749984154442080730571593

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