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  :  $\chi_{77} (2, \cdot )$
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

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

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