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Properties

Label 1-191-191.159-r1-0-0
Degree $1$
Conductor $191$
Sign $0.0589 - 0.998i$
Analytic cond. $20.5258$
Root an. cond. $20.5258$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 191 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0589 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 191 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0589 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(191\)
Sign: $0.0589 - 0.998i$
Analytic conductor: \(20.5258\)
Root analytic conductor: \(20.5258\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{191} (159, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 191,\ (1:\ ),\ 0.0589 - 0.998i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.920674283 - 1.810649968i\)
\(L(\frac12)\) \(\approx\) \(1.920674283 - 1.810649968i\)
\(L(1)\) \(\approx\) \(1.586554894 - 0.4793350004i\)
\(L(1)\) \(\approx\) \(1.586554894 - 0.4793350004i\)

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

−26.724168554497381421989505273212, −25.63274945391628270970791808928, −25.153065232493558751264269576815, −24.224507537096455486847997906982, −23.145315354201503340893056773109, −22.3461221419574670121757781761, −21.942151756681968084703453890495, −20.43852917362531811523952836088, −19.35931928657289330022854907887, −18.41770005189674704935978382798, −17.04596865083896052973285146509, −16.710754083419142139255494576662, −15.09270704406871225170421545338, −14.153358911164576744012808086908, −13.35293498655001463390974236266, −12.49144006141964130017715493359, −11.57617335329851555795352124256, −10.36352268569262535940776319514, −8.88594600751268507545949849169, −7.087881694050661354083207960321, −6.65713574074582544196969517632, −5.87921405293861346487248994838, −4.37559855222420247819270570846, −2.79977572574312970782345564266, −1.81896253642331620572490047274, 0.67119343931311764183098898257, 2.59061595463336058961343957719, 3.77720930034919435576464331753, 4.8794842951373885058690724797, 5.87838775489268036787767755055, 6.642274053581408604720749377096, 8.94833701153959991033022517909, 9.778425319509275074403103353247, 10.71603163986011859730289416915, 11.82701152344140447779895708147, 12.92262000357873565946431419260, 13.65858352115021299745519088741, 14.9914033202259251386238950568, 15.76060055027927510437321308079, 16.763609592519968680725373544205, 17.559146482466328268523617866491, 19.495459461757627209914905017277, 20.05073222588680815066438706951, 21.20952707655071693681206369542, 21.92413017986726254505918839678, 22.484258893244438438510684127337, 23.511060237222040923324957162966, 24.74121626012165562129766947075, 25.41347981174989577024141522459, 26.64985451059527542878490747524

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