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Properties

Label 1-191-191.8-r0-0-0
Degree $1$
Conductor $191$
Sign $-0.714 - 0.699i$
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  + (−0.909 − 0.416i)2-s + (−0.846 − 0.533i)3-s + (0.652 + 0.757i)4-s + (−0.0825 − 0.996i)5-s + (0.546 + 0.837i)6-s + (0.309 − 0.951i)7-s + (−0.277 − 0.960i)8-s + (0.431 + 0.901i)9-s + (−0.340 + 0.940i)10-s + (0.945 + 0.324i)11-s + (−0.148 − 0.988i)12-s + (0.371 − 0.928i)13-s + (−0.677 + 0.735i)14-s + (−0.461 + 0.887i)15-s + (−0.148 + 0.988i)16-s + (0.863 + 0.504i)17-s + ⋯
L(s)  = 1  + (−0.909 − 0.416i)2-s + (−0.846 − 0.533i)3-s + (0.652 + 0.757i)4-s + (−0.0825 − 0.996i)5-s + (0.546 + 0.837i)6-s + (0.309 − 0.951i)7-s + (−0.277 − 0.960i)8-s + (0.431 + 0.901i)9-s + (−0.340 + 0.940i)10-s + (0.945 + 0.324i)11-s + (−0.148 − 0.988i)12-s + (0.371 − 0.928i)13-s + (−0.677 + 0.735i)14-s + (−0.461 + 0.887i)15-s + (−0.148 + 0.988i)16-s + (0.863 + 0.504i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 191 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.714 - 0.699i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 191 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.714 - 0.699i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(191\)
Sign: $-0.714 - 0.699i$
Analytic conductor: \(0.887000\)
Root analytic conductor: \(0.887000\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{191} (8, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 191,\ (0:\ ),\ -0.714 - 0.699i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2255232251 - 0.5530035616i\)
\(L(\frac12)\) \(\approx\) \(0.2255232251 - 0.5530035616i\)
\(L(1)\) \(\approx\) \(0.4841467843 - 0.3720578666i\)
\(L(1)\) \(\approx\) \(0.4841467843 - 0.3720578666i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad191 \( 1 \)
good2 \( 1 + (-0.909 - 0.416i)T \)
3 \( 1 + (-0.846 - 0.533i)T \)
5 \( 1 + (-0.0825 - 0.996i)T \)
7 \( 1 + (0.309 - 0.951i)T \)
11 \( 1 + (0.945 + 0.324i)T \)
13 \( 1 + (0.371 - 0.928i)T \)
17 \( 1 + (0.863 + 0.504i)T \)
19 \( 1 + (-0.340 - 0.940i)T \)
23 \( 1 + (0.828 + 0.560i)T \)
29 \( 1 + (0.894 - 0.446i)T \)
31 \( 1 + (-0.879 - 0.475i)T \)
37 \( 1 + (-0.879 + 0.475i)T \)
41 \( 1 + (-0.677 - 0.735i)T \)
43 \( 1 + (-0.934 - 0.355i)T \)
47 \( 1 + (-0.956 + 0.293i)T \)
53 \( 1 + (-0.0165 + 0.999i)T \)
59 \( 1 + (0.180 - 0.983i)T \)
61 \( 1 + (0.746 + 0.665i)T \)
67 \( 1 + (-0.213 - 0.976i)T \)
71 \( 1 + (0.980 + 0.197i)T \)
73 \( 1 + (-0.574 + 0.818i)T \)
79 \( 1 + (0.894 + 0.446i)T \)
83 \( 1 + (0.828 - 0.560i)T \)
89 \( 1 + (0.0495 + 0.998i)T \)
97 \( 1 + (0.991 + 0.131i)T \)
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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

−27.30632587103930482462781993314, −26.82493288757258675053744154372, −25.621911070187249701729408130624, −24.82806724066315903146517462726, −23.59148656481765440464451585888, −22.816418150671318800506061949477, −21.69470030542296872826630476689, −20.94478537782545391299541870029, −19.30424689333407024167396050754, −18.56282389970045272953707311941, −17.90933322177099407562649724511, −16.69774736787828151571523382285, −16.08558432314811321865255139782, −14.82243311745954522143448845560, −14.4179175263403441487222740215, −12.01713968745420688236886755328, −11.452397714198791244476084706607, −10.48364942342608364924677837824, −9.47730637236395195082122359541, −8.48439375780274544639152701422, −6.88047634925559295443939645541, −6.26603430975736381506122768730, −5.13724242527813654012270413824, −3.3746347315410159663803597776, −1.61798927468352273872269147224, 0.78804027645628955711068249472, 1.61427835146274227831750245217, 3.73054615234870342646233469904, 5.074752923923752616664479046141, 6.570233309369374036893163002928, 7.58656866580128119473983968830, 8.5132510324697587746186072082, 9.84670029437199474613950839823, 10.843585455577074257718463875149, 11.77021586560269674489382060415, 12.66824270836255677822752381968, 13.50093674275108099051970369043, 15.40804973453406981344250460075, 16.62734232134057241398408041113, 17.20327160067093712261797420492, 17.715332083277280266306802911490, 19.124165055352962834240474016477, 19.8802188080814713651045957601, 20.73911265348451856687764020207, 21.8112941891517291054714820453, 23.08685554257599875794551532986, 23.94977418792280470823485209925, 24.91385306221973298213630199711, 25.67989972372877450506102201721, 27.29937285921634103826756746334

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