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Properties

Label 1-191-191.18-r0-0-0
Degree $1$
Conductor $191$
Sign $-0.982 + 0.185i$
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  + (0.863 − 0.504i)2-s + (−0.999 − 0.0330i)3-s + (0.490 − 0.871i)4-s + (−0.677 − 0.735i)5-s + (−0.879 + 0.475i)6-s + (−0.809 + 0.587i)7-s + (−0.0165 − 0.999i)8-s + (0.997 + 0.0660i)9-s + (−0.956 − 0.293i)10-s + (−0.986 + 0.164i)11-s + (−0.518 + 0.854i)12-s + (−0.340 − 0.940i)13-s + (−0.401 + 0.915i)14-s + (0.652 + 0.757i)15-s + (−0.518 − 0.854i)16-s + (−0.627 + 0.778i)17-s + ⋯
L(s)  = 1  + (0.863 − 0.504i)2-s + (−0.999 − 0.0330i)3-s + (0.490 − 0.871i)4-s + (−0.677 − 0.735i)5-s + (−0.879 + 0.475i)6-s + (−0.809 + 0.587i)7-s + (−0.0165 − 0.999i)8-s + (0.997 + 0.0660i)9-s + (−0.956 − 0.293i)10-s + (−0.986 + 0.164i)11-s + (−0.518 + 0.854i)12-s + (−0.340 − 0.940i)13-s + (−0.401 + 0.915i)14-s + (0.652 + 0.757i)15-s + (−0.518 − 0.854i)16-s + (−0.627 + 0.778i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 191 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.982 + 0.185i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 191 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.982 + 0.185i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(191\)
Sign: $-0.982 + 0.185i$
Analytic conductor: \(0.887000\)
Root analytic conductor: \(0.887000\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{191} (18, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 191,\ (0:\ ),\ -0.982 + 0.185i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.04674577782 - 0.4998391699i\)
\(L(\frac12)\) \(\approx\) \(-0.04674577782 - 0.4998391699i\)
\(L(1)\) \(\approx\) \(0.6444137107 - 0.4421234789i\)
\(L(1)\) \(\approx\) \(0.6444137107 - 0.4421234789i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad191 \( 1 \)
good2 \( 1 + (0.863 - 0.504i)T \)
3 \( 1 + (-0.999 - 0.0330i)T \)
5 \( 1 + (-0.677 - 0.735i)T \)
7 \( 1 + (-0.809 + 0.587i)T \)
11 \( 1 + (-0.986 + 0.164i)T \)
13 \( 1 + (-0.340 - 0.940i)T \)
17 \( 1 + (-0.627 + 0.778i)T \)
19 \( 1 + (-0.956 + 0.293i)T \)
23 \( 1 + (-0.574 - 0.818i)T \)
29 \( 1 + (0.922 - 0.386i)T \)
31 \( 1 + (0.245 + 0.969i)T \)
37 \( 1 + (0.245 - 0.969i)T \)
41 \( 1 + (-0.401 - 0.915i)T \)
43 \( 1 + (-0.724 - 0.689i)T \)
47 \( 1 + (-0.461 - 0.887i)T \)
53 \( 1 + (0.894 + 0.446i)T \)
59 \( 1 + (0.371 - 0.928i)T \)
61 \( 1 + (0.0495 + 0.998i)T \)
67 \( 1 + (0.965 + 0.261i)T \)
71 \( 1 + (0.746 - 0.665i)T \)
73 \( 1 + (-0.148 - 0.988i)T \)
79 \( 1 + (0.922 + 0.386i)T \)
83 \( 1 + (-0.574 + 0.818i)T \)
89 \( 1 + (0.180 - 0.983i)T \)
97 \( 1 + (-0.846 - 0.533i)T \)
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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

−27.38224312772764517461182382523, −26.45424081068808396969407427026, −25.84852202325048631275427534441, −24.281572438718266749399128020133, −23.49379242360384288969825830395, −23.07369177705715059923395378601, −22.09075578334766506595550593955, −21.42040820893708081624946855455, −19.99845921218736281152311456610, −18.84741789850412088614955240223, −17.744212524948054140007928852131, −16.59750317885527103929259055056, −15.94249650539892144425208898175, −15.15328744056837602692807361256, −13.755310507383285096863808370973, −12.926694431351001658155947779692, −11.756572368702463755676922448069, −11.07894236246026870782976479672, −9.92013629721106743406251225958, −7.935766910813037589954163703620, −6.865251860617727257428090149812, −6.385941092480407977463581266601, −4.86916187339764158963317405659, −3.98893336611926542368431022776, −2.64992272862093784736866460991, 0.31007086574922158270049826263, 2.23748842809424602529709859738, 3.81020297501615067759339804730, 4.91277713460350287917807933641, 5.750239271502369050599202349515, 6.83740512654420931760848807635, 8.40206883470124067899801917519, 10.08749215317071432329003602071, 10.73802463630180561658651201584, 12.15680075094397195649391808274, 12.52963950027802944064931647411, 13.262329182159197719900437811979, 15.20333133459182358130989164250, 15.68011017034404639590183001660, 16.611923005049283953576873894803, 18.00098034745882882061686991113, 19.110670992800344161894609356373, 19.91299883959756375822252493270, 21.084623506200651644407842979855, 21.911610017879358809530228186203, 22.897384719747895317477795025970, 23.43853582704888782908712484856, 24.35535546263822985681745195073, 25.23494160131975256689480127625, 26.8924626723372461015573231948

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