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Properties

Label 1-191-191.59-r0-0-0
Degree $1$
Conductor $191$
Sign $0.556 - 0.830i$
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  + (0.894 − 0.446i)2-s + (−0.934 + 0.355i)3-s + (0.601 − 0.799i)4-s + (0.945 + 0.324i)5-s + (−0.677 + 0.735i)6-s + (−0.809 − 0.587i)7-s + (0.180 − 0.983i)8-s + (0.746 − 0.665i)9-s + (0.991 − 0.131i)10-s + (0.245 − 0.969i)11-s + (−0.277 + 0.960i)12-s + (−0.627 + 0.778i)13-s + (−0.986 − 0.164i)14-s + (−0.999 + 0.0330i)15-s + (−0.277 − 0.960i)16-s + (0.922 + 0.386i)17-s + ⋯
L(s)  = 1  + (0.894 − 0.446i)2-s + (−0.934 + 0.355i)3-s + (0.601 − 0.799i)4-s + (0.945 + 0.324i)5-s + (−0.677 + 0.735i)6-s + (−0.809 − 0.587i)7-s + (0.180 − 0.983i)8-s + (0.746 − 0.665i)9-s + (0.991 − 0.131i)10-s + (0.245 − 0.969i)11-s + (−0.277 + 0.960i)12-s + (−0.627 + 0.778i)13-s + (−0.986 − 0.164i)14-s + (−0.999 + 0.0330i)15-s + (−0.277 − 0.960i)16-s + (0.922 + 0.386i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 191 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.556 - 0.830i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 191 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.556 - 0.830i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(191\)
Sign: $0.556 - 0.830i$
Analytic conductor: \(0.887000\)
Root analytic conductor: \(0.887000\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{191} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 191,\ (0:\ ),\ 0.556 - 0.830i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.405802733 - 0.7504854722i\)
\(L(\frac12)\) \(\approx\) \(1.405802733 - 0.7504854722i\)
\(L(1)\) \(\approx\) \(1.369396689 - 0.4254427020i\)
\(L(1)\) \(\approx\) \(1.369396689 - 0.4254427020i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad191 \( 1 \)
good2 \( 1 + (0.894 - 0.446i)T \)
3 \( 1 + (-0.934 + 0.355i)T \)
5 \( 1 + (0.945 + 0.324i)T \)
7 \( 1 + (-0.809 - 0.587i)T \)
11 \( 1 + (0.245 - 0.969i)T \)
13 \( 1 + (-0.627 + 0.778i)T \)
17 \( 1 + (0.922 + 0.386i)T \)
19 \( 1 + (0.991 + 0.131i)T \)
23 \( 1 + (0.431 - 0.901i)T \)
29 \( 1 + (-0.340 - 0.940i)T \)
31 \( 1 + (-0.401 - 0.915i)T \)
37 \( 1 + (-0.401 + 0.915i)T \)
41 \( 1 + (-0.986 + 0.164i)T \)
43 \( 1 + (0.490 + 0.871i)T \)
47 \( 1 + (-0.846 - 0.533i)T \)
53 \( 1 + (0.371 + 0.928i)T \)
59 \( 1 + (0.863 + 0.504i)T \)
61 \( 1 + (-0.518 + 0.854i)T \)
67 \( 1 + (-0.973 - 0.229i)T \)
71 \( 1 + (-0.148 + 0.988i)T \)
73 \( 1 + (0.997 + 0.0660i)T \)
79 \( 1 + (-0.340 + 0.940i)T \)
83 \( 1 + (0.431 + 0.901i)T \)
89 \( 1 + (-0.909 + 0.416i)T \)
97 \( 1 + (-0.995 - 0.0990i)T \)
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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

−27.35504022123804885689093393072, −25.67071056636546188092338532577, −25.211118924044578557380711854644, −24.43570829282650528384241982929, −23.28509293651307518323815536483, −22.4381781839227412545107415772, −21.96752300262183101035306957943, −20.87667639114043052855381363044, −19.70652470743666017681192484262, −18.1431455374343189188364628256, −17.45092879575504830572165483706, −16.514115573165951310612783740346, −15.68278995260341411166606001163, −14.44890014945086265674871896546, −13.25101264253444389183039694647, −12.53551785398102019332371576035, −11.93066621402853257690451855702, −10.36192841155722060501659516301, −9.31619128356671553929530433224, −7.47885916570578843343334378630, −6.68407111343714197034307068044, −5.35118469703660672822928316353, −5.23391676267410907074148167557, −3.21875141768139170365567354989, −1.79320508332808408257046108879, 1.19673389461141161818595814772, 2.954755105852580105780385218440, 4.084171645737616358038129135161, 5.41342149537631127386637853148, 6.20152647121304457464395652973, 7.02650687766699332764773211510, 9.56390646260959262021775998078, 10.09809792464592056917351018767, 11.12020904502078475915880553533, 12.08192343015828696611148322541, 13.193664744684854203663805581615, 14.00516087642133017868965325874, 15.06352821355900748516402926876, 16.58606679752643136382700709335, 16.74979772441265321787937703415, 18.46064573364604345661120362091, 19.22970186321733545394123219559, 20.663880728773298580714342056406, 21.43739943042397890763437989051, 22.272626367489968799590202861671, 22.75916834849862405161761773410, 23.92618667490694419953774994060, 24.674762732433409456214861749310, 26.0687978158066418173253627363, 26.95796975571763062332254774985

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