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Properties

Label 1-152-152.123-r1-0-0
Degree $1$
Conductor $152$
Sign $-0.189 + 0.981i$
Analytic cond. $16.3346$
Root an. cond. $16.3346$
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  + (0.173 − 0.984i)3-s + (−0.766 − 0.642i)5-s + (0.5 + 0.866i)7-s + (−0.939 − 0.342i)9-s + (−0.5 + 0.866i)11-s + (−0.173 − 0.984i)13-s + (−0.766 + 0.642i)15-s + (−0.939 + 0.342i)17-s + (0.939 − 0.342i)21-s + (−0.766 + 0.642i)23-s + (0.173 + 0.984i)25-s + (−0.5 + 0.866i)27-s + (0.939 + 0.342i)29-s + (0.5 + 0.866i)31-s + (0.766 + 0.642i)33-s + ⋯
L(s)  = 1  + (0.173 − 0.984i)3-s + (−0.766 − 0.642i)5-s + (0.5 + 0.866i)7-s + (−0.939 − 0.342i)9-s + (−0.5 + 0.866i)11-s + (−0.173 − 0.984i)13-s + (−0.766 + 0.642i)15-s + (−0.939 + 0.342i)17-s + (0.939 − 0.342i)21-s + (−0.766 + 0.642i)23-s + (0.173 + 0.984i)25-s + (−0.5 + 0.866i)27-s + (0.939 + 0.342i)29-s + (0.5 + 0.866i)31-s + (0.766 + 0.642i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.189 + 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.189 + 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(152\)    =    \(2^{3} \cdot 19\)
Sign: $-0.189 + 0.981i$
Analytic conductor: \(16.3346\)
Root analytic conductor: \(16.3346\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{152} (123, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 152,\ (1:\ ),\ -0.189 + 0.981i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2414839419 + 0.2924522296i\)
\(L(\frac12)\) \(\approx\) \(0.2414839419 + 0.2924522296i\)
\(L(1)\) \(\approx\) \(0.7527305401 - 0.1624283731i\)
\(L(1)\) \(\approx\) \(0.7527305401 - 0.1624283731i\)

Euler product

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

−27.18625393788964467353840382494, −26.59156718220474422383133605259, −26.147642148507634109802475013602, −24.40426291241873758535239935307, −23.53574003191404549639461092281, −22.52778117413857845415380106147, −21.61663787259046363379963851001, −20.6208069526124004627803105299, −19.71859283786282459060107753394, −18.71553694113191686026442016650, −17.35783563683054840707737441215, −16.26050946756280903020135604803, −15.55249725932459919411021053472, −14.33601819410455190851620782384, −13.740658194793680770364366028848, −11.74075076755591010989233770301, −10.98841700723072831832882599759, −10.17844512938926349663686283886, −8.7212045586243679600149248609, −7.74632052093046980671132362028, −6.40248760609674966422640528328, −4.66893513315291160746615445157, −3.92837097496660801185203554547, −2.60213632332952489919313630657, −0.13712223536970023163915134475, 1.55105761303832767313984066087, 2.85729215636474106476644348333, 4.64951442008884194680631694855, 5.79932299640629812792895854935, 7.336933399940095682474982335997, 8.16008776577777340915967889696, 9.0211350055999041496902985375, 10.795722796816796772658038377554, 12.23664563436543407936602620755, 12.40845767479779397028022768506, 13.7415043498008068993277916850, 15.126114874615758014160634773215, 15.7455577954289168449330362857, 17.51175635399745248618879950380, 17.95673325391407370955332903571, 19.25881676122127366095728946839, 19.99926766478130158451298948639, 20.90987651627703145998092641224, 22.38292653228089672298947338846, 23.39687009976529636157706932019, 24.223385856791769924152671875604, 24.9559972517268797074835586691, 25.862985305685700873323758273247, 27.27748467122428010710584590274, 28.188702894326166462834254312456

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