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Properties

Label 1-191-191.10-r0-0-0
Degree $1$
Conductor $191$
Sign $-0.425 + 0.905i$
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  + (0.115 − 0.993i)2-s + (−0.768 + 0.639i)3-s + (−0.973 − 0.229i)4-s + (−0.0825 − 0.996i)5-s + (0.546 + 0.837i)6-s + (−0.809 − 0.587i)7-s + (−0.340 + 0.940i)8-s + (0.180 − 0.983i)9-s + (−0.999 − 0.0330i)10-s + (0.945 + 0.324i)11-s + (0.894 − 0.446i)12-s + (−0.846 + 0.533i)13-s + (−0.677 + 0.735i)14-s + (0.701 + 0.712i)15-s + (0.894 + 0.446i)16-s + (−0.995 + 0.0990i)17-s + ⋯
L(s)  = 1  + (0.115 − 0.993i)2-s + (−0.768 + 0.639i)3-s + (−0.973 − 0.229i)4-s + (−0.0825 − 0.996i)5-s + (0.546 + 0.837i)6-s + (−0.809 − 0.587i)7-s + (−0.340 + 0.940i)8-s + (0.180 − 0.983i)9-s + (−0.999 − 0.0330i)10-s + (0.945 + 0.324i)11-s + (0.894 − 0.446i)12-s + (−0.846 + 0.533i)13-s + (−0.677 + 0.735i)14-s + (0.701 + 0.712i)15-s + (0.894 + 0.446i)16-s + (−0.995 + 0.0990i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 191 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.425 + 0.905i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 191 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.425 + 0.905i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(191\)
Sign: $-0.425 + 0.905i$
Analytic conductor: \(0.887000\)
Root analytic conductor: \(0.887000\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{191} (10, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 191,\ (0:\ ),\ -0.425 + 0.905i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.03489085366 - 0.05494964654i\)
\(L(\frac12)\) \(\approx\) \(-0.03489085366 - 0.05494964654i\)
\(L(1)\) \(\approx\) \(0.4517252129 - 0.2561075388i\)
\(L(1)\) \(\approx\) \(0.4517252129 - 0.2561075388i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad191 \( 1 \)
good2 \( 1 + (0.115 - 0.993i)T \)
3 \( 1 + (-0.768 + 0.639i)T \)
5 \( 1 + (-0.0825 - 0.996i)T \)
7 \( 1 + (-0.809 - 0.587i)T \)
11 \( 1 + (0.945 + 0.324i)T \)
13 \( 1 + (-0.846 + 0.533i)T \)
17 \( 1 + (-0.995 + 0.0990i)T \)
19 \( 1 + (-0.999 + 0.0330i)T \)
23 \( 1 + (-0.277 + 0.960i)T \)
29 \( 1 + (-0.461 + 0.887i)T \)
31 \( 1 + (-0.879 - 0.475i)T \)
37 \( 1 + (-0.879 + 0.475i)T \)
41 \( 1 + (-0.677 - 0.735i)T \)
43 \( 1 + (0.965 - 0.261i)T \)
47 \( 1 + (0.601 - 0.799i)T \)
53 \( 1 + (-0.956 + 0.293i)T \)
59 \( 1 + (0.991 - 0.131i)T \)
61 \( 1 + (0.863 - 0.504i)T \)
67 \( 1 + (0.746 + 0.665i)T \)
71 \( 1 + (-0.909 + 0.416i)T \)
73 \( 1 + (-0.0165 - 0.999i)T \)
79 \( 1 + (-0.461 - 0.887i)T \)
83 \( 1 + (-0.277 - 0.960i)T \)
89 \( 1 + (-0.627 - 0.778i)T \)
97 \( 1 + (-0.724 - 0.689i)T \)
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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

−27.528349252217154709393909550, −26.71633001893840214270207444943, −25.59010766031247275808657402336, −24.85719654427112979267684011534, −24.0389603661335772643467425253, −22.83535958806296079886289731704, −22.33533727528598094335353779678, −21.8408982176366824361212295752, −19.46793387770640323309595683163, −18.93003378960173986055794188922, −17.90537915284235601220401591938, −17.1650035281356771148044203933, −16.12427448884269858520420215724, −15.12018068916748840787230418394, −14.1882046697188575515465495851, −13.00218107655745667769262007606, −12.20712976263467938281765764720, −10.911605900197636902032360670625, −9.67042326994800119532622222136, −8.35725944514771718791295047711, −7.01550074522548047329885613937, −6.50563423537670769187395361429, −5.62306790227261749630480304544, −4.105128251158052622761293630404, −2.504902042010556563780901817173, 0.05265408228124922522305228184, 1.71647783758167567938083824061, 3.78106214354707224143345042807, 4.34776946777112147730960646669, 5.49192699968835476369021828928, 6.88522893486786009297916251244, 8.94078084135127732721369609654, 9.4903896756205101922024802008, 10.50985403382711227702965258532, 11.634846551201479397766115022954, 12.41781376668464469636677046172, 13.2591776692374249076318374861, 14.61251368587146051953336396340, 15.89612326453238009226268694435, 17.16943063839047385368650777881, 17.283262687859398501500633248720, 19.06986989473607029079916108045, 19.95203120624621090069029159195, 20.59715934658170684954785648733, 21.88283923039129715117289654129, 22.248330345318896000656587583094, 23.4732104206811246602535178509, 24.05970081018462579255365319475, 25.73616883274332352428344371600, 26.89502622147496040352096908295

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