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Properties

Label 1-191-191.61-r1-0-0
Degree $1$
Conductor $191$
Sign $0.529 + 0.848i$
Analytic cond. $20.5258$
Root an. cond. $20.5258$
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  + (−0.277 − 0.960i)2-s + (0.115 − 0.993i)3-s + (−0.846 + 0.533i)4-s + (0.245 + 0.969i)5-s + (−0.986 + 0.164i)6-s + (0.809 − 0.587i)7-s + (0.746 + 0.665i)8-s + (−0.973 − 0.229i)9-s + (0.863 − 0.504i)10-s + (−0.546 − 0.837i)11-s + (0.431 + 0.901i)12-s + (−0.909 + 0.416i)13-s + (−0.789 − 0.614i)14-s + (0.991 − 0.131i)15-s + (0.431 − 0.901i)16-s + (−0.0165 + 0.999i)17-s + ⋯
L(s)  = 1  + (−0.277 − 0.960i)2-s + (0.115 − 0.993i)3-s + (−0.846 + 0.533i)4-s + (0.245 + 0.969i)5-s + (−0.986 + 0.164i)6-s + (0.809 − 0.587i)7-s + (0.746 + 0.665i)8-s + (−0.973 − 0.229i)9-s + (0.863 − 0.504i)10-s + (−0.546 − 0.837i)11-s + (0.431 + 0.901i)12-s + (−0.909 + 0.416i)13-s + (−0.789 − 0.614i)14-s + (0.991 − 0.131i)15-s + (0.431 − 0.901i)16-s + (−0.0165 + 0.999i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 191 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.529 + 0.848i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 191 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.529 + 0.848i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(191\)
Sign: $0.529 + 0.848i$
Analytic conductor: \(20.5258\)
Root analytic conductor: \(20.5258\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{191} (61, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 191,\ (1:\ ),\ 0.529 + 0.848i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3262140893 + 0.1808910193i\)
\(L(\frac12)\) \(\approx\) \(0.3262140893 + 0.1808910193i\)
\(L(1)\) \(\approx\) \(0.6342015074 - 0.3639323103i\)
\(L(1)\) \(\approx\) \(0.6342015074 - 0.3639323103i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad191 \( 1 \)
good2 \( 1 + (-0.277 - 0.960i)T \)
3 \( 1 + (0.115 - 0.993i)T \)
5 \( 1 + (0.245 + 0.969i)T \)
7 \( 1 + (0.809 - 0.587i)T \)
11 \( 1 + (-0.546 - 0.837i)T \)
13 \( 1 + (-0.909 + 0.416i)T \)
17 \( 1 + (-0.0165 + 0.999i)T \)
19 \( 1 + (-0.863 - 0.504i)T \)
23 \( 1 + (-0.213 + 0.976i)T \)
29 \( 1 + (-0.180 + 0.983i)T \)
31 \( 1 + (0.0825 + 0.996i)T \)
37 \( 1 + (0.0825 - 0.996i)T \)
41 \( 1 + (-0.789 + 0.614i)T \)
43 \( 1 + (-0.461 - 0.887i)T \)
47 \( 1 + (0.627 - 0.778i)T \)
53 \( 1 + (-0.0495 + 0.998i)T \)
59 \( 1 + (-0.518 + 0.854i)T \)
61 \( 1 + (0.574 - 0.818i)T \)
67 \( 1 + (0.601 + 0.799i)T \)
71 \( 1 + (-0.828 - 0.560i)T \)
73 \( 1 + (-0.965 - 0.261i)T \)
79 \( 1 + (0.180 + 0.983i)T \)
83 \( 1 + (0.213 + 0.976i)T \)
89 \( 1 + (0.148 + 0.988i)T \)
97 \( 1 + (0.922 + 0.386i)T \)
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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

−26.78474250520017053577554153835, −25.60506954867483221876471070020, −24.94397819003697977366466251529, −24.115286503802026573724489068812, −22.924262493033182209909869257025, −22.03667097751369753711075252755, −20.873081432147444741461838856976, −20.264344848314885615066677273961, −18.78519409228736201310875872442, −17.55995261297388458420574555203, −16.99378447924894040424064584058, −15.94357978268127051874866031486, −15.12046627947903802674774067406, −14.41586741012509311417488556966, −13.104660194661413485558969881741, −11.866115198187394583687382256878, −10.239768483391927444353696569035, −9.54235519132418413035444410451, −8.502313655704161232051731716756, −7.766243195821188438459902823732, −5.93078087205160874960826962236, −4.89679658256933085286474531823, −4.5119207479111358752524123527, −2.22881108255677205020712474523, −0.136518405746514144049390648948, 1.518985030159497207601192347549, 2.48448688511932310431069512462, 3.68382227625954922152443058716, 5.36560760758098769534193376493, 6.92753748435690492124512585195, 7.842242015630494372233969356652, 8.84959912474664409499228429485, 10.43128792797077913984664726164, 11.04254269473819245054994965220, 12.04983168275845006855584345421, 13.2470960754829908777065538439, 13.98975351982806609135314721045, 14.80969323382009155357680303297, 16.96624639383181543828907476742, 17.623147269236715370638510578, 18.47343212822314472440028853755, 19.30258501990499454943772981388, 19.98232626619474836079510851495, 21.4257748118358219655173961454, 21.87971798881685715892985565085, 23.43265300729203158078676631972, 23.76925874533883838012112036896, 25.28890830847842945613502283295, 26.34446639688355929710284880616, 26.83802415308290932817170173373

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