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Properties

Label 1-191-191.20-r0-0-0
Degree $1$
Conductor $191$
Sign $0.344 + 0.938i$
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  + (0.965 − 0.261i)2-s + (−0.0165 + 0.999i)3-s + (0.863 − 0.504i)4-s + (−0.401 + 0.915i)5-s + (0.245 + 0.969i)6-s + (0.309 + 0.951i)7-s + (0.701 − 0.712i)8-s + (−0.999 − 0.0330i)9-s + (−0.148 + 0.988i)10-s + (−0.0825 − 0.996i)11-s + (0.490 + 0.871i)12-s + (−0.574 + 0.818i)13-s + (0.546 + 0.837i)14-s + (−0.909 − 0.416i)15-s + (0.490 − 0.871i)16-s + (0.431 + 0.901i)17-s + ⋯
L(s)  = 1  + (0.965 − 0.261i)2-s + (−0.0165 + 0.999i)3-s + (0.863 − 0.504i)4-s + (−0.401 + 0.915i)5-s + (0.245 + 0.969i)6-s + (0.309 + 0.951i)7-s + (0.701 − 0.712i)8-s + (−0.999 − 0.0330i)9-s + (−0.148 + 0.988i)10-s + (−0.0825 − 0.996i)11-s + (0.490 + 0.871i)12-s + (−0.574 + 0.818i)13-s + (0.546 + 0.837i)14-s + (−0.909 − 0.416i)15-s + (0.490 − 0.871i)16-s + (0.431 + 0.901i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 191 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.344 + 0.938i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 191 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.344 + 0.938i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(191\)
Sign: $0.344 + 0.938i$
Analytic conductor: \(0.887000\)
Root analytic conductor: \(0.887000\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{191} (20, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 191,\ (0:\ ),\ 0.344 + 0.938i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.541813315 + 1.076078798i\)
\(L(\frac12)\) \(\approx\) \(1.541813315 + 1.076078798i\)
\(L(1)\) \(\approx\) \(1.567054584 + 0.5712801652i\)
\(L(1)\) \(\approx\) \(1.567054584 + 0.5712801652i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad191 \( 1 \)
good2 \( 1 + (0.965 - 0.261i)T \)
3 \( 1 + (-0.0165 + 0.999i)T \)
5 \( 1 + (-0.401 + 0.915i)T \)
7 \( 1 + (0.309 + 0.951i)T \)
11 \( 1 + (-0.0825 - 0.996i)T \)
13 \( 1 + (-0.574 + 0.818i)T \)
17 \( 1 + (0.431 + 0.901i)T \)
19 \( 1 + (-0.148 - 0.988i)T \)
23 \( 1 + (-0.461 + 0.887i)T \)
29 \( 1 + (0.980 - 0.197i)T \)
31 \( 1 + (0.789 + 0.614i)T \)
37 \( 1 + (0.789 - 0.614i)T \)
41 \( 1 + (0.546 - 0.837i)T \)
43 \( 1 + (0.371 - 0.928i)T \)
47 \( 1 + (-0.518 + 0.854i)T \)
53 \( 1 + (-0.973 - 0.229i)T \)
59 \( 1 + (0.828 - 0.560i)T \)
61 \( 1 + (-0.724 - 0.689i)T \)
67 \( 1 + (0.991 + 0.131i)T \)
71 \( 1 + (-0.934 + 0.355i)T \)
73 \( 1 + (0.652 - 0.757i)T \)
79 \( 1 + (0.980 + 0.197i)T \)
83 \( 1 + (-0.461 - 0.887i)T \)
89 \( 1 + (-0.768 + 0.639i)T \)
97 \( 1 + (-0.277 + 0.960i)T \)
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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

−26.77836938948882319277348347465, −25.410705267105807359467489357937, −24.81117652308123598600007651572, −24.02583833396261874010987244455, −23.07826964778590578332293131038, −22.75924457657434391109464301359, −20.914283836100002013117515800437, −20.27002664722598467198999122969, −19.66424426561883397747017349739, −18.00501129044138405286587455985, −17.05232521935747614927404513273, −16.312713828048370959078864436872, −14.91866047466654956751596245391, −14.04837561170404813714410390164, −13.00828764003772284840271896289, −12.37274477820892029301418212668, −11.56633321231793594904669441094, −10.07463784873262275246002416564, −8.03897190397722115281792578992, −7.70102284705225212718499889625, −6.507135289919510215027609350526, −5.15333864664110393716950805537, −4.28397457655470068532942858933, −2.6998070527066288040015754110, −1.20761510665225515952573150706, 2.3851610231740359428376114167, 3.30259940524366094063355660981, 4.3947229242935420068977368027, 5.56847424939483176207078056757, 6.492919157369891974061037728820, 8.10200401862953220616781202633, 9.49995780413879432661961276077, 10.71586187573578184362972900879, 11.38905248679984711565449428612, 12.206858185283852127577958135900, 13.93305368012933029717352115, 14.54495149279110478185768357247, 15.49128702221937574432545214241, 16.03885611085999653803175078756, 17.50117702991102743899879986700, 19.13505284648244066442895150947, 19.51597182966655102669085899945, 21.11120986693585535074183513138, 21.74244131157487732458862521200, 22.09125618599321619001777206336, 23.35499686440924283934760899582, 24.11475968354394995419863054330, 25.4245887949331066183370750710, 26.31802983218988594012741223993, 27.32165152261553957055691920384

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