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Properties

Label 1-191-191.3-r0-0-0
Degree $1$
Conductor $191$
Sign $-0.999 + 0.0375i$
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  + (0.652 − 0.757i)2-s + (0.431 − 0.901i)3-s + (−0.148 − 0.988i)4-s + (−0.986 − 0.164i)5-s + (−0.401 − 0.915i)6-s + (−0.809 + 0.587i)7-s + (−0.846 − 0.533i)8-s + (−0.627 − 0.778i)9-s + (−0.768 + 0.639i)10-s + (0.789 − 0.614i)11-s + (−0.956 − 0.293i)12-s + (−0.724 + 0.689i)13-s + (−0.0825 + 0.996i)14-s + (−0.574 + 0.818i)15-s + (−0.956 + 0.293i)16-s + (0.490 − 0.871i)17-s + ⋯
L(s)  = 1  + (0.652 − 0.757i)2-s + (0.431 − 0.901i)3-s + (−0.148 − 0.988i)4-s + (−0.986 − 0.164i)5-s + (−0.401 − 0.915i)6-s + (−0.809 + 0.587i)7-s + (−0.846 − 0.533i)8-s + (−0.627 − 0.778i)9-s + (−0.768 + 0.639i)10-s + (0.789 − 0.614i)11-s + (−0.956 − 0.293i)12-s + (−0.724 + 0.689i)13-s + (−0.0825 + 0.996i)14-s + (−0.574 + 0.818i)15-s + (−0.956 + 0.293i)16-s + (0.490 − 0.871i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 191 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0375i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 191 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0375i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(191\)
Sign: $-0.999 + 0.0375i$
Analytic conductor: \(0.887000\)
Root analytic conductor: \(0.887000\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{191} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 191,\ (0:\ ),\ -0.999 + 0.0375i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.02196938026 - 1.168992024i\)
\(L(\frac12)\) \(\approx\) \(0.02196938026 - 1.168992024i\)
\(L(1)\) \(\approx\) \(0.7094769798 - 0.9320499082i\)
\(L(1)\) \(\approx\) \(0.7094769798 - 0.9320499082i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad191 \( 1 \)
good2 \( 1 + (0.652 - 0.757i)T \)
3 \( 1 + (0.431 - 0.901i)T \)
5 \( 1 + (-0.986 - 0.164i)T \)
7 \( 1 + (-0.809 + 0.587i)T \)
11 \( 1 + (0.789 - 0.614i)T \)
13 \( 1 + (-0.724 + 0.689i)T \)
17 \( 1 + (0.490 - 0.871i)T \)
19 \( 1 + (-0.768 - 0.639i)T \)
23 \( 1 + (0.371 - 0.928i)T \)
29 \( 1 + (0.601 + 0.799i)T \)
31 \( 1 + (0.546 - 0.837i)T \)
37 \( 1 + (0.546 + 0.837i)T \)
41 \( 1 + (-0.0825 - 0.996i)T \)
43 \( 1 + (0.746 - 0.665i)T \)
47 \( 1 + (0.828 + 0.560i)T \)
53 \( 1 + (-0.999 + 0.0330i)T \)
59 \( 1 + (-0.934 + 0.355i)T \)
61 \( 1 + (0.115 - 0.993i)T \)
67 \( 1 + (-0.909 - 0.416i)T \)
71 \( 1 + (0.922 - 0.386i)T \)
73 \( 1 + (-0.340 + 0.940i)T \)
79 \( 1 + (0.601 - 0.799i)T \)
83 \( 1 + (0.371 + 0.928i)T \)
89 \( 1 + (-0.995 - 0.0990i)T \)
97 \( 1 + (0.965 - 0.261i)T \)
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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

−27.14581035799068688691275810686, −26.65802107864524721999401445400, −25.57185700667473041501894540569, −24.96969396362240326947574541122, −23.40523838617891111563264592271, −22.93889114515416449980398608490, −22.091574611758988002566066331431, −21.06739438455647079073564053707, −19.861206448407827627974564643673, −19.37103195429202361831875479857, −17.372126969308752073448165278078, −16.668883501412127391440499130201, −15.69576835681443499969851481486, −14.976353511564227941818893048282, −14.26839099120976203268917968865, −12.9068914935583675895030934442, −12.00327695785146101105762154100, −10.59687804654779824521918159787, −9.482094060230971516066951522810, −8.18389875093624164676296331327, −7.37411308637611977703823799741, −6.10291277396910494418157583350, −4.58166896784064588250264957585, −3.86041262467590452497260993055, −2.97471975794695575464675167117, 0.70524335356331246346701873202, 2.45768276104761718696666681257, 3.32715317526747756883369066399, 4.60836291150518884450499840451, 6.20461472784924118368224226176, 7.05773964576801333343793340663, 8.675560644719504886487966829780, 9.40393216374782914956376204975, 11.12766642972180392248130075170, 12.12168452999441176423673009352, 12.48064929771299009375545628596, 13.73314650206124045808221711861, 14.60636208691698479474921808962, 15.58694512577895051449723850891, 16.88024897327806795736057955176, 18.6545053952548726462118731400, 19.06768596205033596196082969002, 19.73036902042457429144925481546, 20.66577036238840991720982318591, 22.00155224317789160285700859975, 22.762416872388763392356463865156, 23.79359411351074476909897333217, 24.40726887477260574261033733822, 25.385890228316602933754708348838, 26.75970379239985260384415338248

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