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Properties

Label 1-191-191.56-r1-0-0
Degree $1$
Conductor $191$
Sign $0.271 - 0.962i$
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  + (0.490 + 0.871i)2-s + (0.997 − 0.0660i)3-s + (−0.518 + 0.854i)4-s + (−0.0825 − 0.996i)5-s + (0.546 + 0.837i)6-s + (−0.309 − 0.951i)7-s + (−0.999 − 0.0330i)8-s + (0.991 − 0.131i)9-s + (0.828 − 0.560i)10-s + (−0.945 − 0.324i)11-s + (−0.461 + 0.887i)12-s + (−0.768 − 0.639i)13-s + (0.677 − 0.735i)14-s + (−0.148 − 0.988i)15-s + (−0.461 − 0.887i)16-s + (−0.213 + 0.976i)17-s + ⋯
L(s)  = 1  + (0.490 + 0.871i)2-s + (0.997 − 0.0660i)3-s + (−0.518 + 0.854i)4-s + (−0.0825 − 0.996i)5-s + (0.546 + 0.837i)6-s + (−0.309 − 0.951i)7-s + (−0.999 − 0.0330i)8-s + (0.991 − 0.131i)9-s + (0.828 − 0.560i)10-s + (−0.945 − 0.324i)11-s + (−0.461 + 0.887i)12-s + (−0.768 − 0.639i)13-s + (0.677 − 0.735i)14-s + (−0.148 − 0.988i)15-s + (−0.461 − 0.887i)16-s + (−0.213 + 0.976i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 191 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.271 - 0.962i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 191 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.271 - 0.962i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(191\)
Sign: $0.271 - 0.962i$
Analytic conductor: \(20.5258\)
Root analytic conductor: \(20.5258\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{191} (56, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 191,\ (1:\ ),\ 0.271 - 0.962i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.429966600 - 1.082148793i\)
\(L(\frac12)\) \(\approx\) \(1.429966600 - 1.082148793i\)
\(L(1)\) \(\approx\) \(1.371457844 + 0.04120186408i\)
\(L(1)\) \(\approx\) \(1.371457844 + 0.04120186408i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad191 \( 1 \)
good2 \( 1 + (0.490 + 0.871i)T \)
3 \( 1 + (0.997 - 0.0660i)T \)
5 \( 1 + (-0.0825 - 0.996i)T \)
7 \( 1 + (-0.309 - 0.951i)T \)
11 \( 1 + (-0.945 - 0.324i)T \)
13 \( 1 + (-0.768 - 0.639i)T \)
17 \( 1 + (-0.213 + 0.976i)T \)
19 \( 1 + (-0.828 - 0.560i)T \)
23 \( 1 + (-0.340 - 0.940i)T \)
29 \( 1 + (-0.701 - 0.712i)T \)
31 \( 1 + (0.879 + 0.475i)T \)
37 \( 1 + (0.879 - 0.475i)T \)
41 \( 1 + (0.677 + 0.735i)T \)
43 \( 1 + (0.0495 - 0.998i)T \)
47 \( 1 + (0.574 + 0.818i)T \)
53 \( 1 + (-0.601 + 0.799i)T \)
59 \( 1 + (-0.724 + 0.689i)T \)
61 \( 1 + (0.995 + 0.0990i)T \)
67 \( 1 + (0.863 - 0.504i)T \)
71 \( 1 + (-0.115 - 0.993i)T \)
73 \( 1 + (0.956 + 0.293i)T \)
79 \( 1 + (0.701 - 0.712i)T \)
83 \( 1 + (0.340 - 0.940i)T \)
89 \( 1 + (0.934 - 0.355i)T \)
97 \( 1 + (0.431 - 0.901i)T \)
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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

−27.04857356043248248894985022207, −26.163741964798980649826774597713, −25.244509057462572516588013760026, −24.11282868419966172516131279160, −23.023215588067615596812218549836, −21.98250053925981424885683156749, −21.40700788930449291012103775340, −20.407089508116404231087453950836, −19.277648397089426510992896195383, −18.78439249495046213975242669451, −17.99176188262040898096703992967, −15.81274531190037383040828227888, −15.04046314787347373476060199484, −14.31925168441512075262948220344, −13.30302238058738936137128715450, −12.31580271552102412714269212487, −11.21011903242526403241434884293, −9.95135449146036943059243207548, −9.392398421007864110793508195147, −7.94896576914375165000888397933, −6.62704798456035708414358532810, −5.16251518816500419082384855071, −3.824210074496333229483661533887, −2.63448919855698925689490406680, −2.17891915013651850901862112295, 0.431460765335281033172440605111, 2.58350495471119228872981104734, 3.98033221929054563634051255706, 4.74287452191436002883408940402, 6.24037804303749586010356328219, 7.64262309700360063375018400168, 8.151857415532037496551779195210, 9.247693487813826132902324199849, 10.461649762968262720886512903444, 12.56743497218451717834469944594, 13.00817836672296072757469634767, 13.84156518257207261651155792730, 14.99441019776735741413978479606, 15.79910466976851167668717563960, 16.778622236028446840438209678162, 17.6372652486889276783938378118, 19.11227895371880469450999668520, 20.10686732707177586035247683245, 20.915273583180004999973257594270, 21.81095435598179514535430134494, 23.22638631832388306340222822120, 24.02271281570526551818434420439, 24.62154841794658130153792022015, 25.66927491012379736849416199683, 26.4498846382428910178572651843

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