↑

Properties

Label 1-191-191.12-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

Related objects

Downloads

Learn more

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: $\chi_{191} (12, \cdot )$
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 \( 1 + (-0.340 + 0.940i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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