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Properties

Label 1-152-152.147-r0-0-0
Degree $1$
Conductor $152$
Sign $0.837 - 0.546i$
Analytic cond. $0.705885$
Root an. cond. $0.705885$
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  + (0.939 + 0.342i)3-s + (−0.173 − 0.984i)5-s + (0.5 − 0.866i)7-s + (0.766 + 0.642i)9-s + (−0.5 − 0.866i)11-s + (−0.939 + 0.342i)13-s + (0.173 − 0.984i)15-s + (0.766 − 0.642i)17-s + (0.766 − 0.642i)21-s + (−0.173 + 0.984i)23-s + (−0.939 + 0.342i)25-s + (0.5 + 0.866i)27-s + (0.766 + 0.642i)29-s + (−0.5 + 0.866i)31-s + (−0.173 − 0.984i)33-s + ⋯
L(s)  = 1  + (0.939 + 0.342i)3-s + (−0.173 − 0.984i)5-s + (0.5 − 0.866i)7-s + (0.766 + 0.642i)9-s + (−0.5 − 0.866i)11-s + (−0.939 + 0.342i)13-s + (0.173 − 0.984i)15-s + (0.766 − 0.642i)17-s + (0.766 − 0.642i)21-s + (−0.173 + 0.984i)23-s + (−0.939 + 0.342i)25-s + (0.5 + 0.866i)27-s + (0.766 + 0.642i)29-s + (−0.5 + 0.866i)31-s + (−0.173 − 0.984i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.837 - 0.546i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.837 - 0.546i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(152\)    =    \(2^{3} \cdot 19\)
Sign: $0.837 - 0.546i$
Analytic conductor: \(0.705885\)
Root analytic conductor: \(0.705885\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{152} (147, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 152,\ (0:\ ),\ 0.837 - 0.546i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.396285079 - 0.4149399742i\)
\(L(\frac12)\) \(\approx\) \(1.396285079 - 0.4149399742i\)
\(L(1)\) \(\approx\) \(1.335400298 - 0.2013831622i\)
\(L(1)\) \(\approx\) \(1.335400298 - 0.2013831622i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
19 \( 1 \)
good3 \( 1 + (0.939 + 0.342i)T \)
5 \( 1 + (-0.173 - 0.984i)T \)
7 \( 1 + (0.5 - 0.866i)T \)
11 \( 1 + (-0.5 - 0.866i)T \)
13 \( 1 + (-0.939 + 0.342i)T \)
17 \( 1 + (0.766 - 0.642i)T \)
23 \( 1 + (-0.173 + 0.984i)T \)
29 \( 1 + (0.766 + 0.642i)T \)
31 \( 1 + (-0.5 + 0.866i)T \)
37 \( 1 + T \)
41 \( 1 + (0.939 + 0.342i)T \)
43 \( 1 + (0.173 + 0.984i)T \)
47 \( 1 + (-0.766 - 0.642i)T \)
53 \( 1 + (0.173 - 0.984i)T \)
59 \( 1 + (-0.766 + 0.642i)T \)
61 \( 1 + (-0.173 + 0.984i)T \)
67 \( 1 + (-0.766 - 0.642i)T \)
71 \( 1 + (0.173 + 0.984i)T \)
73 \( 1 + (-0.939 - 0.342i)T \)
79 \( 1 + (-0.939 - 0.342i)T \)
83 \( 1 + (-0.5 + 0.866i)T \)
89 \( 1 + (0.939 - 0.342i)T \)
97 \( 1 + (-0.766 + 0.642i)T \)
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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

−27.98298696816431936375672376493, −26.978697603880591692326119531189, −26.07321847344765185594400067231, −25.30465896821260982648891008067, −24.3961367739646140624495739313, −23.29532251499954044616567275997, −22.13205141565561841040410174314, −21.200523517550596009667112755854, −20.14829219914961348822154202943, −19.061085693605547144613885723531, −18.40810844638146378898162840238, −17.47809328751560618450430012922, −15.61077336984668002172231267023, −14.84516793571871951728088231871, −14.333411406366456123560121754479, −12.79507286297725796010829541034, −11.99821990262742616311577342770, −10.46732079486291750202566504249, −9.509295630809339176235650477061, −8.06787890992943151544083064129, −7.44644720570656509557654921824, −6.06231719971608492384646833913, −4.41170980427325221984893278688, −2.84918195357682700172732783351, −2.117776602265091698264608046685, 1.34125820925626771986023456664, 3.05776981882503139109707960349, 4.34401962392888593519006030294, 5.25610113608670389328868501179, 7.38056589884516385839024001175, 8.107608383399949856041930914516, 9.22343242638528407613780766973, 10.21383624415523299396562447930, 11.55434174244921041546794789334, 12.91584017434545518495613688544, 13.837589813652002847901603710094, 14.65565492794661249863655460405, 16.099601217144904037124472740550, 16.59402474935488989094847640285, 17.98788566820790665989690861082, 19.45887711116813656809601482339, 19.95783538920235903615069954188, 21.087626484457330298279271885625, 21.546999952138126073370903643253, 23.327796300160552090824201492390, 24.16443464045302750370724272574, 24.93845225603238755147984185543, 26.081661695890388293250964170191, 27.13443504913387426671853891740, 27.49310075252568818880382144725

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