Properties

Degree 1
Conductor $ 7 \cdot 11 $
Sign $-0.624 + 0.781i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.978 − 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

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (-0.624 + 0.781i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 77 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (-0.624 + 0.781i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(77\)    =    \(7 \cdot 11\)
\( \varepsilon \)  =  $-0.624 + 0.781i$
motivic weight  =  \(0\)
character  :  $\chi_{77} (59, \cdot )$
Sato-Tate  :  $\mu(30)$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 77,\ (1:\ ),\ -0.624 + 0.781i)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(-0.1151294862 - 0.2393518603i\)
\(L(\frac12,\chi)\)  \(\approx\)  \(-0.1151294862 - 0.2393518603i\)
\(L(\chi,1)\)  \(\approx\)  \(0.4163337123 - 0.2930042850i\)
\(L(1,\chi)\)  \(\approx\)  \(0.4163337123 - 0.2930042850i\)

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

−31.83755547201695287710628229696, −30.56001036603616863009962130670, −29.31545233808577118241199399070, −27.98482079520296381984303566631, −27.44974489474998936928093201073, −26.18570263733253070329129896992, −25.97562911304579747886301244422, −24.23703166289592352321642089298, −23.05637418238028760706132997749, −21.74170608501481308397777562504, −20.62217673033795086002125674035, −19.45120460443461052727469973693, −18.64949369748002627808749641023, −17.091857387319626984340266293809, −16.244987272577396810237675201176, −15.102285681231770018981871289330, −14.41497923742296551430684475224, −11.8833287636447174090420440532, −10.9010291535842373775791005266, −9.93776023597150132771095085249, −8.716998706615419695995020535211, −7.48370415622714958129309334044, −6.05018598536805890984049220299, −4.13023738987600501882676080644, −2.54180706652128659531365323953, 0.1723106175390166654072120125, 1.57656629180911972297913377485, 3.32781154291160252868423602809, 5.72749105879624206628640543294, 7.42081899179604931759192157120, 8.07367609385282539720023899955, 9.281947150487250093204122569404, 10.94219587603111149695659479081, 12.2067221991621554213436275569, 12.82508133774253752710043451296, 14.73737554947154126119879663085, 16.15969993241783000134310447664, 17.19745133609336529148914605076, 18.230509718397648101674782944648, 19.378848354933436454134534477394, 19.99098481277878155356646500976, 21.11807003492511941507130367933, 23.02483407273507932829057173517, 24.1028760660918591707044247605, 25.02025033940084872183394979925, 25.87257146249783772340919687025, 27.422379125901819680294003206916, 28.009847560651749131602559006685, 29.32500934469724953555833412130, 29.98477089830405885374150233153

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