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Properties

Label 1-191-191.38-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: $\chi_{191} (38, \cdot )$
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

−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