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Properties

Label 1-191-191.6-r0-0-0
Degree $1$
Conductor $191$
Sign $0.967 + 0.251i$
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  + (0.945 + 0.324i)2-s + (−0.401 − 0.915i)3-s + (0.789 + 0.614i)4-s + (0.789 + 0.614i)5-s + (−0.0825 − 0.996i)6-s + 7-s + (0.546 + 0.837i)8-s + (−0.677 + 0.735i)9-s + (0.546 + 0.837i)10-s + (−0.879 − 0.475i)11-s + (0.245 − 0.969i)12-s + (−0.401 + 0.915i)13-s + (0.945 + 0.324i)14-s + (0.245 − 0.969i)15-s + (0.245 + 0.969i)16-s + (−0.986 − 0.164i)17-s + ⋯
L(s)  = 1  + (0.945 + 0.324i)2-s + (−0.401 − 0.915i)3-s + (0.789 + 0.614i)4-s + (0.789 + 0.614i)5-s + (−0.0825 − 0.996i)6-s + 7-s + (0.546 + 0.837i)8-s + (−0.677 + 0.735i)9-s + (0.546 + 0.837i)10-s + (−0.879 − 0.475i)11-s + (0.245 − 0.969i)12-s + (−0.401 + 0.915i)13-s + (0.945 + 0.324i)14-s + (0.245 − 0.969i)15-s + (0.245 + 0.969i)16-s + (−0.986 − 0.164i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 191 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.967 + 0.251i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 191 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.967 + 0.251i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(191\)
Sign: $0.967 + 0.251i$
Analytic conductor: \(0.887000\)
Root analytic conductor: \(0.887000\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{191} (6, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 191,\ (0:\ ),\ 0.967 + 0.251i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.024118600 + 0.2581752032i\)
\(L(\frac12)\) \(\approx\) \(2.024118600 + 0.2581752032i\)
\(L(1)\) \(\approx\) \(1.767050770 + 0.1393126390i\)
\(L(1)\) \(\approx\) \(1.767050770 + 0.1393126390i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad191 \( 1 \)
good2 \( 1 + (0.945 + 0.324i)T \)
3 \( 1 + (-0.401 - 0.915i)T \)
5 \( 1 + (0.789 + 0.614i)T \)
7 \( 1 + T \)
11 \( 1 + (-0.879 - 0.475i)T \)
13 \( 1 + (-0.401 + 0.915i)T \)
17 \( 1 + (-0.986 - 0.164i)T \)
19 \( 1 + (0.546 - 0.837i)T \)
23 \( 1 + (0.546 - 0.837i)T \)
29 \( 1 + (0.245 - 0.969i)T \)
31 \( 1 + (-0.677 + 0.735i)T \)
37 \( 1 + (-0.677 - 0.735i)T \)
41 \( 1 + (0.945 - 0.324i)T \)
43 \( 1 + (-0.0825 - 0.996i)T \)
47 \( 1 + (-0.879 - 0.475i)T \)
53 \( 1 + (-0.879 - 0.475i)T \)
59 \( 1 + (-0.677 + 0.735i)T \)
61 \( 1 + (-0.986 + 0.164i)T \)
67 \( 1 + (-0.986 + 0.164i)T \)
71 \( 1 + (0.945 - 0.324i)T \)
73 \( 1 + (-0.879 + 0.475i)T \)
79 \( 1 + (0.245 + 0.969i)T \)
83 \( 1 + (0.546 + 0.837i)T \)
89 \( 1 + (-0.0825 + 0.996i)T \)
97 \( 1 + (-0.677 - 0.735i)T \)
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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

−27.39427009480609161058955163476, −25.98697586578323514502252977622, −24.917166845760643146738377346281, −24.07407779065734292298355941080, −23.0928724933161647555986540138, −22.10820495393941658653397214935, −21.34257378289363479450724966368, −20.61008260111689742322950595328, −20.07813435357450192343810139189, −18.10554428855971580966078638382, −17.32675558208770934831586273761, −16.14079649496346054292121265150, −15.22001990608950633183721478536, −14.41109479838884826824084362602, −13.222241045754352511177649535748, −12.30673133358046150480756827390, −11.1105720318971693015701569873, −10.35154425633485967399190920124, −9.34997029164285388774712029015, −7.77552087627880122972335809125, −6.02418241997895002406835102382, −5.11166810014186696784884020970, −4.63871561673016983618027150975, −3.03239828616179372901615426210, −1.600403041408494192939664564868, 1.92993522628657074723230533496, 2.70650483109228699323805644015, 4.73066179895439732328932974317, 5.576100946690277122789201136738, 6.71589050001732590446111746073, 7.41471252357329752776971611639, 8.71298476366557433179721450138, 10.80731303476686934685351319440, 11.30310131372625841939888766676, 12.51030367218050640537445391813, 13.66548162040549231524239224271, 14.00871862901651182074181703839, 15.18048323930770428438628514453, 16.5145647371751866957393608474, 17.55350854463443853396251818439, 18.16214393569047542163512826430, 19.40497500398796902945337471681, 20.78859338828282336248442401323, 21.607999607153191905632590195740, 22.42194488148338693506613114784, 23.44787578662097513416218463823, 24.365237927616863983405259568963, 24.70964973824208018106601844717, 26.02944039180359681807148370221, 26.72550283727425472972734769244

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