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Properties

Label 1-191-191.60-r0-0-0
Degree $1$
Conductor $191$
Sign $-0.298 - 0.954i$
Analytic cond. $0.887000$
Root an. cond. $0.887000$
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  + (0.431 − 0.901i)2-s + (0.894 + 0.446i)3-s + (−0.627 − 0.778i)4-s + (0.546 − 0.837i)5-s + (0.789 − 0.614i)6-s + (−0.809 − 0.587i)7-s + (−0.973 + 0.229i)8-s + (0.601 + 0.799i)9-s + (−0.518 − 0.854i)10-s + (−0.677 − 0.735i)11-s + (−0.213 − 0.976i)12-s + (−0.148 − 0.988i)13-s + (−0.879 + 0.475i)14-s + (0.863 − 0.504i)15-s + (−0.213 + 0.976i)16-s + (0.997 + 0.0660i)17-s + ⋯
L(s)  = 1  + (0.431 − 0.901i)2-s + (0.894 + 0.446i)3-s + (−0.627 − 0.778i)4-s + (0.546 − 0.837i)5-s + (0.789 − 0.614i)6-s + (−0.809 − 0.587i)7-s + (−0.973 + 0.229i)8-s + (0.601 + 0.799i)9-s + (−0.518 − 0.854i)10-s + (−0.677 − 0.735i)11-s + (−0.213 − 0.976i)12-s + (−0.148 − 0.988i)13-s + (−0.879 + 0.475i)14-s + (0.863 − 0.504i)15-s + (−0.213 + 0.976i)16-s + (0.997 + 0.0660i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 191 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.298 - 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 191 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.298 - 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(191\)
Sign: $-0.298 - 0.954i$
Analytic conductor: \(0.887000\)
Root analytic conductor: \(0.887000\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{191} (60, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 191,\ (0:\ ),\ -0.298 - 0.954i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.010973118 - 1.375482288i\)
\(L(\frac12)\) \(\approx\) \(1.010973118 - 1.375482288i\)
\(L(1)\) \(\approx\) \(1.245759017 - 0.8788820206i\)
\(L(1)\) \(\approx\) \(1.245759017 - 0.8788820206i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad191 \( 1 \)
good2 \( 1 + (0.431 - 0.901i)T \)
3 \( 1 + (0.894 + 0.446i)T \)
5 \( 1 + (0.546 - 0.837i)T \)
7 \( 1 + (-0.809 - 0.587i)T \)
11 \( 1 + (-0.677 - 0.735i)T \)
13 \( 1 + (-0.148 - 0.988i)T \)
17 \( 1 + (0.997 + 0.0660i)T \)
19 \( 1 + (-0.518 + 0.854i)T \)
23 \( 1 + (0.652 + 0.757i)T \)
29 \( 1 + (0.746 + 0.665i)T \)
31 \( 1 + (0.945 - 0.324i)T \)
37 \( 1 + (0.945 + 0.324i)T \)
41 \( 1 + (-0.879 - 0.475i)T \)
43 \( 1 + (-0.340 - 0.940i)T \)
47 \( 1 + (-0.909 + 0.416i)T \)
53 \( 1 + (0.980 + 0.197i)T \)
59 \( 1 + (-0.574 + 0.818i)T \)
61 \( 1 + (-0.768 + 0.639i)T \)
67 \( 1 + (-0.846 - 0.533i)T \)
71 \( 1 + (-0.724 + 0.689i)T \)
73 \( 1 + (0.490 + 0.871i)T \)
79 \( 1 + (0.746 - 0.665i)T \)
83 \( 1 + (0.652 - 0.757i)T \)
89 \( 1 + (0.828 - 0.560i)T \)
97 \( 1 + (-0.0165 + 0.999i)T \)
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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

−26.62909917339262100303800773102, −26.22760817159355280916223083107, −25.3824894683131822444187003757, −24.89044894924277541388576116961, −23.54156114852607292605076858820, −22.87195195329439651412078503157, −21.59290755971090208879716665277, −21.09767169008413962575186972242, −19.40201884182018161548675088963, −18.59797022064217178425784262161, −17.864806318710581504062680503901, −16.55145422258233905763587711793, −15.301855219232784295591911777029, −14.78701681478110372598652977256, −13.74043607728535455426823189656, −12.99374872531945838949256612937, −12.00648284471161588904557616173, −9.97805184338835203041077671365, −9.2127197630517503578830272527, −7.99780396852409636166921046316, −6.84130824249503035389965153355, −6.33937195958707337791498658305, −4.72348748044987288400617602530, −3.16730146039172424803765186399, −2.40372505849830651790691976865, 1.16634746163574460724745705723, 2.7791566052165687154352365303, 3.589095456885115582664125984528, 4.89108987385385781454313582451, 5.89880406383952719305363333373, 7.939871769711516260284336110755, 8.95251430945743907441990541952, 10.103405527873164232208264120770, 10.406606206221778624780845724909, 12.26387261688105362637478517612, 13.2511203750739997694226237693, 13.63087158878730286526644524269, 14.85965767616057976600474530347, 16.001582987014814408777833092276, 17.03206002487965237085025155873, 18.55336761932974873310235208884, 19.463492008672701132178232661694, 20.24494349148508612420996344128, 21.03185102302001629576851889407, 21.63212086513429281541497228838, 22.86213617650955882582420770063, 23.81778622154107300315627955353, 25.01845869915220910545971996363, 25.79867115468596325395400432790, 27.08142817878786073510990096771

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