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Properties

Label 1-191-191.54-r0-0-0
Degree $1$
Conductor $191$
Sign $0.957 + 0.287i$
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  + (0.180 − 0.983i)2-s + (−0.461 + 0.887i)3-s + (−0.934 − 0.355i)4-s + (0.546 + 0.837i)5-s + (0.789 + 0.614i)6-s + (0.309 − 0.951i)7-s + (−0.518 + 0.854i)8-s + (−0.574 − 0.818i)9-s + (0.922 − 0.386i)10-s + (−0.677 + 0.735i)11-s + (0.746 − 0.665i)12-s + (0.894 + 0.446i)13-s + (−0.879 − 0.475i)14-s + (−0.995 + 0.0990i)15-s + (0.746 + 0.665i)16-s + (0.371 + 0.928i)17-s + ⋯
L(s)  = 1  + (0.180 − 0.983i)2-s + (−0.461 + 0.887i)3-s + (−0.934 − 0.355i)4-s + (0.546 + 0.837i)5-s + (0.789 + 0.614i)6-s + (0.309 − 0.951i)7-s + (−0.518 + 0.854i)8-s + (−0.574 − 0.818i)9-s + (0.922 − 0.386i)10-s + (−0.677 + 0.735i)11-s + (0.746 − 0.665i)12-s + (0.894 + 0.446i)13-s + (−0.879 − 0.475i)14-s + (−0.995 + 0.0990i)15-s + (0.746 + 0.665i)16-s + (0.371 + 0.928i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 191 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.957 + 0.287i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 191 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.957 + 0.287i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(191\)
Sign: $0.957 + 0.287i$
Analytic conductor: \(0.887000\)
Root analytic conductor: \(0.887000\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{191} (54, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 191,\ (0:\ ),\ 0.957 + 0.287i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.031194092 + 0.1513902700i\)
\(L(\frac12)\) \(\approx\) \(1.031194092 + 0.1513902700i\)
\(L(1)\) \(\approx\) \(0.9951494103 - 0.04864400897i\)
\(L(1)\) \(\approx\) \(0.9951494103 - 0.04864400897i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad191 \( 1 \)
good2 \( 1 + (0.180 - 0.983i)T \)
3 \( 1 + (-0.461 + 0.887i)T \)
5 \( 1 + (0.546 + 0.837i)T \)
7 \( 1 + (0.309 - 0.951i)T \)
11 \( 1 + (-0.677 + 0.735i)T \)
13 \( 1 + (0.894 + 0.446i)T \)
17 \( 1 + (0.371 + 0.928i)T \)
19 \( 1 + (0.922 + 0.386i)T \)
23 \( 1 + (-0.973 + 0.229i)T \)
29 \( 1 + (0.863 + 0.504i)T \)
31 \( 1 + (0.945 + 0.324i)T \)
37 \( 1 + (0.945 - 0.324i)T \)
41 \( 1 + (-0.879 + 0.475i)T \)
43 \( 1 + (-0.999 - 0.0330i)T \)
47 \( 1 + (0.115 - 0.993i)T \)
53 \( 1 + (-0.909 - 0.416i)T \)
59 \( 1 + (-0.0165 + 0.999i)T \)
61 \( 1 + (0.997 + 0.0660i)T \)
67 \( 1 + (-0.768 - 0.639i)T \)
71 \( 1 + (0.431 - 0.901i)T \)
73 \( 1 + (0.980 + 0.197i)T \)
79 \( 1 + (0.863 - 0.504i)T \)
83 \( 1 + (-0.973 - 0.229i)T \)
89 \( 1 + (-0.277 + 0.960i)T \)
97 \( 1 + (-0.956 - 0.293i)T \)
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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

−26.976541066294556533411176970220, −25.55938050536737888891563558097, −25.03373427920025953874516684868, −24.26565874096837326283727010615, −23.605214863429218883489292475304, −22.471341902306218006330511945610, −21.59131001955351544113460315968, −20.49037274214439853791438320437, −18.79256464928640998874953934850, −18.17688773977222221176468903294, −17.463929032889143580882165794060, −16.24642736456233189031613158903, −15.71659250282948578051728222082, −13.983826730616606502136370455440, −13.48297717315862975741650568157, −12.47287782286980402022538938693, −11.55419412013561163806794095071, −9.75186806634563099788098376263, −8.423832844744754947279742223317, −7.99765482891678226175594802562, −6.36103849388225606103063849264, −5.62435961029131815424122814927, −4.91296918558600747008361354189, −2.77269512648983091706112101707, −0.92795349250675786889097961025, 1.55817629428616381544696188148, 3.19383647886835583441325906990, 4.11006233161812422276751233074, 5.2670718145264029902377736594, 6.44929787305574043553208347881, 8.179203147463971191469606721902, 9.797587308269634728258495954003, 10.23990947765546296415345972908, 11.022482483626366774808170403883, 12.0241058786097702186872475641, 13.48379742990210577197668470071, 14.24788987619549924629443104177, 15.22772799460669618278918042500, 16.617372423196824513413252061250, 17.80074093277540362903297780207, 18.23179759691960628226314155257, 19.74664107475337539436196036207, 20.69771167990857545546673030531, 21.3426368265570268725998579649, 22.18683265209489794871340688514, 23.23137140361585360129417062849, 23.562143961413710441038351534634, 25.7127527170854172933420433179, 26.48315417730020171510379726662, 27.106682242542369708848826579390

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