Properties

Degree 1
Conductor 191
Sign $0.997 - 0.0694i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.879 + 0.475i)2-s + (−0.986 + 0.164i)3-s + (0.546 − 0.837i)4-s + (0.546 − 0.837i)5-s + (0.789 − 0.614i)6-s − 7-s + (−0.0825 + 0.996i)8-s + (0.945 − 0.324i)9-s + (−0.0825 + 0.996i)10-s + (0.677 + 0.735i)11-s + (−0.401 + 0.915i)12-s + (−0.986 − 0.164i)13-s + (0.879 − 0.475i)14-s + (−0.401 + 0.915i)15-s + (−0.401 − 0.915i)16-s + (0.245 + 0.969i)17-s + ⋯
L(s,χ)  = 1  + (−0.879 + 0.475i)2-s + (−0.986 + 0.164i)3-s + (0.546 − 0.837i)4-s + (0.546 − 0.837i)5-s + (0.789 − 0.614i)6-s − 7-s + (−0.0825 + 0.996i)8-s + (0.945 − 0.324i)9-s + (−0.0825 + 0.996i)10-s + (0.677 + 0.735i)11-s + (−0.401 + 0.915i)12-s + (−0.986 − 0.164i)13-s + (0.879 − 0.475i)14-s + (−0.401 + 0.915i)15-s + (−0.401 − 0.915i)16-s + (0.245 + 0.969i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 191 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (0.997 - 0.0694i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 191 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (0.997 - 0.0694i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(191\)
\( \varepsilon \)  =  $0.997 - 0.0694i$
motivic weight  =  \(0\)
character  :  $\chi_{191} (38, \cdot )$
Sato-Tate  :  $\mu(38)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 191,\ (1:\ ),\ 0.997 - 0.0694i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.7045176695 + 0.02448258065i$
$L(\frac12,\chi)$  $\approx$  $0.7045176695 + 0.02448258065i$
$L(\chi,1)$  $\approx$  0.5483560928 + 0.04997955892i
$L(1,\chi)$  $\approx$  0.5483560928 + 0.04997955892i

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

−27.02730255849460512618279475693, −26.078637575061960124028547569662, −25.173100037851635605273415459885, −24.149329267189339605524633128032, −22.63722213690972116677561928894, −22.062844311387826580306152310089, −21.41654554404632136141699033500, −19.73340745109871548138612749236, −19.07322490225944130526024639866, −18.1690442317472591208616036207, −17.323359427342638263237471854714, −16.52078322267450869601540189195, −15.592504826427883755465506854744, −13.89695789813055815128086186855, −12.76993523374848801244207889087, −11.71573379418651335479884772089, −10.94234192406425175332972880930, −9.87651081348478887586068315469, −9.23370890442596925611513886067, −7.20109600405323050625903612730, −6.8099974737046021109736445739, −5.53076355778702977967783733655, −3.57345843605252053862532287122, −2.372869009030445003584726262805, −0.73467285811364623537948755334, 0.596520117588029384108629251597, 1.94363427460343244094808208502, 4.30436468409439395488646019638, 5.61761432214091441139996278465, 6.323568225209976093328677313202, 7.44387343134180170145595127629, 8.98010645890248344700925437263, 9.874055330018853909617856201, 10.43038121632434971527281699771, 12.15594535356515094742348010635, 12.62193311717899571657933975132, 14.36823016930052554673118906843, 15.55071547725256178004195742065, 16.56417893582651978517034450444, 17.01736488804925309814079500994, 17.802511839731022232596480262978, 19.02326204772427808523219170249, 19.93867900337947943386156312968, 21.02718454486304096982519887988, 22.27747600108118834425782521037, 23.09572500715494628231277112392, 24.23794852476136451418661073110, 24.9392858259476423788743693172, 25.85871216235262067150326241347, 26.9931125277780183851346457057

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