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Properties

Label 1-191-191.16-r0-0-0
Degree $1$
Conductor $191$
Sign $-0.182 - 0.983i$
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  + (0.0495 − 0.998i)2-s + (−0.956 + 0.293i)3-s + (−0.995 − 0.0990i)4-s + (−0.401 + 0.915i)5-s + (0.245 + 0.969i)6-s + (−0.809 + 0.587i)7-s + (−0.148 + 0.988i)8-s + (0.828 − 0.560i)9-s + (0.894 + 0.446i)10-s + (−0.0825 − 0.996i)11-s + (0.980 − 0.197i)12-s + (−0.0165 − 0.999i)13-s + (0.546 + 0.837i)14-s + (0.115 − 0.993i)15-s + (0.980 + 0.197i)16-s + (0.180 − 0.983i)17-s + ⋯
L(s)  = 1  + (0.0495 − 0.998i)2-s + (−0.956 + 0.293i)3-s + (−0.995 − 0.0990i)4-s + (−0.401 + 0.915i)5-s + (0.245 + 0.969i)6-s + (−0.809 + 0.587i)7-s + (−0.148 + 0.988i)8-s + (0.828 − 0.560i)9-s + (0.894 + 0.446i)10-s + (−0.0825 − 0.996i)11-s + (0.980 − 0.197i)12-s + (−0.0165 − 0.999i)13-s + (0.546 + 0.837i)14-s + (0.115 − 0.993i)15-s + (0.980 + 0.197i)16-s + (0.180 − 0.983i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 191 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.182 - 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 191 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.182 - 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(191\)
Sign: $-0.182 - 0.983i$
Analytic conductor: \(0.887000\)
Root analytic conductor: \(0.887000\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{191} (16, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 191,\ (0:\ ),\ -0.182 - 0.983i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3471000612 - 0.4173447786i\)
\(L(\frac12)\) \(\approx\) \(0.3471000612 - 0.4173447786i\)
\(L(1)\) \(\approx\) \(0.5687730918 - 0.2542191156i\)
\(L(1)\) \(\approx\) \(0.5687730918 - 0.2542191156i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad191 \( 1 \)
good2 \( 1 + (0.0495 - 0.998i)T \)
3 \( 1 + (-0.956 + 0.293i)T \)
5 \( 1 + (-0.401 + 0.915i)T \)
7 \( 1 + (-0.809 + 0.587i)T \)
11 \( 1 + (-0.0825 - 0.996i)T \)
13 \( 1 + (-0.0165 - 0.999i)T \)
17 \( 1 + (0.180 - 0.983i)T \)
19 \( 1 + (0.894 - 0.446i)T \)
23 \( 1 + (0.701 + 0.712i)T \)
29 \( 1 + (-0.909 - 0.416i)T \)
31 \( 1 + (0.789 + 0.614i)T \)
37 \( 1 + (0.789 - 0.614i)T \)
41 \( 1 + (0.546 - 0.837i)T \)
43 \( 1 + (-0.846 + 0.533i)T \)
47 \( 1 + (0.922 - 0.386i)T \)
53 \( 1 + (-0.518 + 0.854i)T \)
59 \( 1 + (-0.277 - 0.960i)T \)
61 \( 1 + (0.431 - 0.901i)T \)
67 \( 1 + (-0.724 - 0.689i)T \)
71 \( 1 + (0.965 + 0.261i)T \)
73 \( 1 + (-0.973 + 0.229i)T \)
79 \( 1 + (-0.909 + 0.416i)T \)
83 \( 1 + (0.701 - 0.712i)T \)
89 \( 1 + (0.997 - 0.0660i)T \)
97 \( 1 + (-0.340 - 0.940i)T \)
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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

−27.285910174037329551792923067, −26.36904274208246635395212631832, −25.29042449135212131094289954167, −24.29333788316829814569072598967, −23.60350245016371659583474118049, −22.93987157991328248899352103175, −22.09068404673370080948802885629, −20.74213855295985610269733117258, −19.39884996429770072938522979139, −18.50633258334249539473467091698, −17.24035699000821740610586975883, −16.69312026478457159806102417389, −16.05664878791436466706786399015, −14.90258055334021256849423399068, −13.38357734666010281998001665757, −12.76360423461811490624644629113, −11.84552529643079713089311612366, −10.18804119985913235236320732120, −9.27235380240236728660779832761, −7.82131525095904071606242675136, −6.97764016142915562211744523821, −5.98442181811029671179419136723, −4.76267099752089971971168555888, −4.00686783101289209216302048843, −1.186412743456330417040166475470, 0.57698622518312639374232913241, 2.856860852820394263478763690895, 3.53141906430230114228997749415, 5.17201308166406609755679975227, 6.04660554214368783536394241382, 7.51905274915509510129620917357, 9.19160866181831692529473868344, 10.0910112599257411633266236762, 11.10095783760695213499519158326, 11.678254812684888418353046709393, 12.75203343224199500584370786440, 13.81572018315623927184703328861, 15.259712900994128772650107069655, 16.04336857774236257794776166620, 17.44306291201311826719562873136, 18.4227159445313882613094927952, 18.94111961517389682207952240395, 20.08001533812513134603050585387, 21.39475409343520154833662445201, 22.12223838702791642817140831452, 22.741164454673142961981938248233, 23.40501382536759936837122332932, 24.850680923535974833452566283507, 26.408511613521543387181911513549, 27.002546624400663419537901648989

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