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Properties

Label 1-191-191.186-r1-0-0
Degree $1$
Conductor $191$
Sign $0.997 + 0.0694i$
Analytic cond. $20.5258$
Root an. cond. $20.5258$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(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)  = 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

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

Invariants

Degree: \(1\)
Conductor: \(191\)
Sign: $0.997 + 0.0694i$
Analytic conductor: \(20.5258\)
Root analytic conductor: \(20.5258\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{191} (186, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 191,\ (1:\ ),\ 0.997 + 0.0694i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7045176695 + 0.02448258065i\)
\(L(\frac12)\) \(\approx\) \(0.7045176695 + 0.02448258065i\)
\(L(1)\) \(\approx\) \(0.5483560928 - 0.04997955892i\)
\(L(1)\) \(\approx\) \(0.5483560928 - 0.04997955892i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad191 \( 1 \)
good2 \( 1 + (-0.879 - 0.475i)T \)
3 \( 1 + (-0.986 - 0.164i)T \)
5 \( 1 + (0.546 + 0.837i)T \)
7 \( 1 - T \)
11 \( 1 + (0.677 - 0.735i)T \)
13 \( 1 + (-0.986 + 0.164i)T \)
17 \( 1 + (0.245 - 0.969i)T \)
19 \( 1 + (0.0825 - 0.996i)T \)
23 \( 1 + (-0.0825 + 0.996i)T \)
29 \( 1 + (0.401 + 0.915i)T \)
31 \( 1 + (-0.945 - 0.324i)T \)
37 \( 1 + (-0.945 + 0.324i)T \)
41 \( 1 + (0.879 - 0.475i)T \)
43 \( 1 + (0.789 + 0.614i)T \)
47 \( 1 + (0.677 - 0.735i)T \)
53 \( 1 + (0.677 - 0.735i)T \)
59 \( 1 + (0.945 + 0.324i)T \)
61 \( 1 + (-0.245 - 0.969i)T \)
67 \( 1 + (0.245 + 0.969i)T \)
71 \( 1 + (0.879 - 0.475i)T \)
73 \( 1 + (0.677 + 0.735i)T \)
79 \( 1 + (-0.401 + 0.915i)T \)
83 \( 1 + (0.0825 + 0.996i)T \)
89 \( 1 + (-0.789 + 0.614i)T \)
97 \( 1 + (0.945 - 0.324i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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