Properties

Label 1-152-152.101-r0-0-0
Degree $1$
Conductor $152$
Sign $-0.188 - 0.982i$
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  + (−0.766 − 0.642i)3-s + (0.939 − 0.342i)5-s + (−0.5 − 0.866i)7-s + (0.173 + 0.984i)9-s + (0.5 − 0.866i)11-s + (−0.766 + 0.642i)13-s + (−0.939 − 0.342i)15-s + (0.173 − 0.984i)17-s + (−0.173 + 0.984i)21-s + (−0.939 − 0.342i)23-s + (0.766 − 0.642i)25-s + (0.5 − 0.866i)27-s + (−0.173 − 0.984i)29-s + (−0.5 − 0.866i)31-s + (−0.939 + 0.342i)33-s + ⋯
L(s)  = 1  + (−0.766 − 0.642i)3-s + (0.939 − 0.342i)5-s + (−0.5 − 0.866i)7-s + (0.173 + 0.984i)9-s + (0.5 − 0.866i)11-s + (−0.766 + 0.642i)13-s + (−0.939 − 0.342i)15-s + (0.173 − 0.984i)17-s + (−0.173 + 0.984i)21-s + (−0.939 − 0.342i)23-s + (0.766 − 0.642i)25-s + (0.5 − 0.866i)27-s + (−0.173 − 0.984i)29-s + (−0.5 − 0.866i)31-s + (−0.939 + 0.342i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.188 - 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.188 - 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(152\)    =    \(2^{3} \cdot 19\)
Sign: $-0.188 - 0.982i$
Analytic conductor: \(0.705885\)
Root analytic conductor: \(0.705885\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{152} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 152,\ (0:\ ),\ -0.188 - 0.982i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5546247993 - 0.6712391888i\)
\(L(\frac12)\) \(\approx\) \(0.5546247993 - 0.6712391888i\)
\(L(1)\) \(\approx\) \(0.7977802953 - 0.3878781122i\)
\(L(1)\) \(\approx\) \(0.7977802953 - 0.3878781122i\)

Euler product

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

−28.20611790958829530871539883978, −27.63587015378261283240740339099, −26.23363653363231728138825613760, −25.53362441037814091151842698, −24.49089950479604570453361306782, −23.12968429467007709441283070676, −22.04253416829264833765301299837, −21.919727632276209430613495182749, −20.62641651391601476194032962552, −19.36725642974778087124089660428, −17.945150958792768617306840228192, −17.512689907585373611278239676762, −16.34189043323290356311112552343, −15.20052826553794468606563999343, −14.46612754270219564956381227898, −12.73839791558192166162559565653, −12.1178571410542125217784041445, −10.59090077246945154683585887653, −9.87285866445738178245054919007, −8.96633241650142575608887463632, −6.99877147699615075330693097579, −5.93202846309373085223735776689, −5.13964671642916839912680042555, −3.526751898540040130291726725, −1.97495632397080903605018970567, 0.83595704519343706261788627905, 2.34881049249478035831683816746, 4.302519759197259301828826653495, 5.662826318568666248229703819387, 6.54172291463389107039495880384, 7.60815863476138183383663690578, 9.25468995976436701894438247147, 10.2385991730668460600359921896, 11.44274257716122545896274532059, 12.497499766323358026197702876317, 13.622864332286052368534353371464, 14.11030777682482614086006293413, 16.250998196212757804589041812376, 16.78853837207329485066037237375, 17.62164150155039232806067466488, 18.75957090822318265557700779429, 19.69183664775121147536557814673, 20.940110709201329220576663478721, 22.09785598641606574324180920382, 22.735779724273175143929517591950, 24.12003173451698370867538166498, 24.51219110687196804359929389642, 25.72940262989066845782320630634, 26.78889592074783653570079928326, 27.945446122901085076673855558294

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