Properties

Degree 1
Conductor $ 7 \cdot 11 $
Sign $-0.180 + 0.983i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

Learn more about

Normalization:  

Dirichlet series

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

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (-0.180 + 0.983i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 77 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (-0.180 + 0.983i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(77\)    =    \(7 \cdot 11\)
\( \varepsilon \)  =  $-0.180 + 0.983i$
motivic weight  =  \(0\)
character  :  $\chi_{77} (18, \cdot )$
Sato-Tate  :  $\mu(30)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 77,\ (1:\ ),\ -0.180 + 0.983i)$
$L(\chi,\frac{1}{2})$  $\approx$  $1.982918219 + 2.379334336i$
$L(\frac12,\chi)$  $\approx$  $1.982918219 + 2.379334336i$
$L(\chi,1)$  $\approx$  1.696385461 + 1.085870459i
$L(1,\chi)$  $\approx$  1.696385461 + 1.085870459i

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

−30.640000669415047221970406184334, −29.71394914743614022719603663535, −28.8860515475806935832205476048, −28.08917903649864519079976769213, −25.85220895833616546940043963926, −25.005230466069619255015344324396, −24.053782746180898172652466184240, −23.37897919364672324150503283050, −21.96694840846171679351245196897, −20.99520476730592580262047285656, −19.841040384554893281840409553987, −18.816947794674268378341872549152, −17.26522616805924533129502392873, −16.321278573799957178976912195443, −14.50106186081454741788685633719, −13.69748866685658727089241720852, −12.60173641468955730433382555328, −11.91213041517135064800365682424, −10.31157489868910884833406952140, −8.55336577132694936918794215858, −6.90278313893999812356310499912, −5.88322495292309912502040971885, −4.591119461609039246781800379805, −2.53177169170176872069930537087, −1.275101530942613042093447257060, 2.571801943120830993350832107060, 3.76483299075739646201872476114, 5.30447187517561070304911984402, 6.20324202906335945289341455566, 7.888566441547385247794402871486, 9.87203944276821932589631634301, 10.74141316426056731724146373116, 12.04311386732827519126813852915, 13.58034143929872775995276260861, 14.63266616501654574015981683679, 15.372174663556649106104635819268, 16.66793940159920873800222687653, 17.68430056652823366648016589234, 19.56476019048333802267959089894, 20.92705709004308937705103916806, 21.550266206311522164246874808112, 22.58977457213228975525401910056, 23.26985669014996403883312299059, 25.03537565035009650253969964705, 25.72687945384453555267381649576, 26.833257315904888625904705012128, 28.14069691409525923645728832461, 29.51785668545654427934563653919, 30.16158967301073086765824309242, 31.66751722092153046954857677333

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