Properties

Degree 1
Conductor $ 7 \cdot 11 $
Sign $0.999 - 0.0413i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

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

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (0.999 - 0.0413i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 77 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (0.999 - 0.0413i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(77\)    =    \(7 \cdot 11\)
\( \varepsilon \)  =  $0.999 - 0.0413i$
motivic weight  =  \(0\)
character  :  $\chi_{77} (39, \cdot )$
Sato-Tate  :  $\mu(30)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 77,\ (1:\ ),\ 0.999 - 0.0413i)$
$L(\chi,\frac{1}{2})$  $\approx$  $2.251266370 - 0.04659604814i$
$L(\frac12,\chi)$  $\approx$  $2.251266370 - 0.04659604814i$
$L(\chi,1)$  $\approx$  1.498402727 + 0.1413716346i
$L(1,\chi)$  $\approx$  1.498402727 + 0.1413716346i

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

−30.75037878403591246700101997783, −30.21496415072832001247827802635, −28.70998989422990517666257497504, −28.08581015808625623097830085642, −26.51354596499764812775969804940, −26.16878695402520888675146099834, −24.6559135843507593997065260164, −22.95635564991275309795003744775, −21.889451028496555182096312143017, −21.213844427744360310328603944934, −20.29831562889206396352988518972, −19.090509776141324821779712668635, −18.05742444410418972854212299957, −16.66342617994356019511955865763, −14.98785685841571180234051172474, −13.979494611147309883417695654083, −13.19398400678703895436873270603, −11.40028719708086034170650677385, −10.283909304573664965284519269452, −9.48104522007468944025756757631, −8.29329981995462965908975166992, −5.95897081106166522923643912631, −4.41551453043681888151884653747, −3.108943844379974919248144134763, −1.81470676096514541535317368083, 1.16516552451578813067856332630, 3.251295101468902437200171426505, 5.24001688395738564355787655872, 6.39459928550965748784628097927, 7.67443176994234681418999322979, 8.81058179120644084335464802873, 9.80558248802568372072154476534, 12.19879006266049391013034606139, 13.50841151388209643882894299324, 13.86033624399808846697747541059, 15.28771555214577840372637713511, 16.53631980727868584508690602585, 17.9208637954264177159000449005, 18.35108684442796904129676650686, 20.05443061112682363341189051477, 21.17498266785572947498203039862, 22.572637763272856117082712151629, 23.75338362835300443949938053628, 24.767504529970548512347284194666, 25.37349466850366118527242014407, 26.22963162036450171505915222196, 27.56475618503838174118037578745, 28.97345397555375924623370938685, 30.110345402649798439892923014649, 31.207210327987978488668842690853

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