Properties

Degree 1
Conductor 191
Sign $0.967 + 0.251i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

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Normalization:  

Dirichlet series

L(χ,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,χ)  = 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

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

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(191\)
\( \varepsilon \)  =  $0.967 + 0.251i$
motivic weight  =  \(0\)
character  :  $\chi_{191} (6, \cdot )$
Sato-Tate  :  $\mu(19)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 191,\ (0:\ ),\ 0.967 + 0.251i)$
$L(\chi,\frac{1}{2})$  $\approx$  $2.024118600 + 0.2581752032i$
$L(\frac12,\chi)$  $\approx$  $2.024118600 + 0.2581752032i$
$L(\chi,1)$  $\approx$  1.767050770 + 0.1393126390i
$L(1,\chi)$  $\approx$  1.767050770 + 0.1393126390i

Euler product

\[\begin{aligned} L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]
\[\begin{aligned} L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]

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