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Properties

Label 1-1161-1161.988-r1-0-0
Degree $1$
Conductor $1161$
Sign $-0.802 - 0.597i$
Analytic cond. $124.766$
Root an. cond. $124.766$
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)2-s + (−0.939 + 0.342i)4-s + (−0.766 − 0.642i)5-s + (0.939 + 0.342i)7-s + (0.5 + 0.866i)8-s + (−0.5 + 0.866i)10-s + (0.766 − 0.642i)11-s + (0.173 − 0.984i)13-s + (0.173 − 0.984i)14-s + (0.766 − 0.642i)16-s + (−0.5 + 0.866i)17-s + (0.5 + 0.866i)19-s + (0.939 + 0.342i)20-s + (−0.766 − 0.642i)22-s + (−0.939 + 0.342i)23-s + ⋯
L(s)  = 1  + (−0.173 − 0.984i)2-s + (−0.939 + 0.342i)4-s + (−0.766 − 0.642i)5-s + (0.939 + 0.342i)7-s + (0.5 + 0.866i)8-s + (−0.5 + 0.866i)10-s + (0.766 − 0.642i)11-s + (0.173 − 0.984i)13-s + (0.173 − 0.984i)14-s + (0.766 − 0.642i)16-s + (−0.5 + 0.866i)17-s + (0.5 + 0.866i)19-s + (0.939 + 0.342i)20-s + (−0.766 − 0.642i)22-s + (−0.939 + 0.342i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1161 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.802 - 0.597i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1161 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.802 - 0.597i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1161\)    =    \(3^{3} \cdot 43\)
Sign: $-0.802 - 0.597i$
Analytic conductor: \(124.766\)
Root analytic conductor: \(124.766\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1161} (988, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1161,\ (1:\ ),\ -0.802 - 0.597i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4652585898 - 1.404071392i\)
\(L(\frac12)\) \(\approx\) \(0.4652585898 - 1.404071392i\)
\(L(1)\) \(\approx\) \(0.7408561236 - 0.5163494005i\)
\(L(1)\) \(\approx\) \(0.7408561236 - 0.5163494005i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
43 \( 1 \)
good2 \( 1 + (-0.173 - 0.984i)T \)
5 \( 1 + (-0.766 - 0.642i)T \)
7 \( 1 + (0.939 + 0.342i)T \)
11 \( 1 + (0.766 - 0.642i)T \)
13 \( 1 + (0.173 - 0.984i)T \)
17 \( 1 + (-0.5 + 0.866i)T \)
19 \( 1 + (0.5 + 0.866i)T \)
23 \( 1 + (-0.939 + 0.342i)T \)
29 \( 1 + (-0.173 - 0.984i)T \)
31 \( 1 + (-0.939 + 0.342i)T \)
37 \( 1 + (0.5 - 0.866i)T \)
41 \( 1 + (0.173 - 0.984i)T \)
47 \( 1 + (-0.939 - 0.342i)T \)
53 \( 1 + T \)
59 \( 1 + (0.766 + 0.642i)T \)
61 \( 1 + (0.939 + 0.342i)T \)
67 \( 1 + (0.173 - 0.984i)T \)
71 \( 1 + (0.5 - 0.866i)T \)
73 \( 1 + (0.5 + 0.866i)T \)
79 \( 1 + (0.173 + 0.984i)T \)
83 \( 1 + (0.173 + 0.984i)T \)
89 \( 1 + (0.5 + 0.866i)T \)
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

−21.77576885793794879794503098213, −20.31245002079651811750321464700, −19.86762123461246000151623824657, −18.81955501531539671133610366832, −18.10967996062882699263479011410, −17.670105389028137481779046621892, −16.56780077723085211450727379354, −16.055813678349318072482828736908, −15.079731383113376357963544714467, −14.48937468923259389824652176125, −14.02542993204756969764940370514, −12.995212943051447024485644650004, −11.65899526774739081043888570322, −11.35668373609950904522578915090, −10.1752654221540969346792700040, −9.2792908291619600353297469569, −8.50148453514850504385811186784, −7.55398314011099001792506052696, −7.02193960140742281652420984989, −6.37612246479251516352352165130, −4.95302343830020174253213262358, −4.427684167020658198166004920405, −3.59571895697630046411425840095, −2.05194842252595070076037006940, −0.84523859202368117767120159116, 0.44001940093300744665295080681, 1.33738109034855626605193158256, 2.23158685338158588994750384292, 3.76382854137755983313756070852, 3.89226615609379169508397753001, 5.194224426090830400916024304031, 5.82626108438122928522065811891, 7.56684176044694055571157383042, 8.23891235104334134457101441833, 8.70147527388018998244223223297, 9.67446766494313243735484065312, 10.742354151103099159935067046482, 11.36238546416503374473558998236, 12.042785017144442595309075919953, 12.66484505995527065675725433402, 13.58980897934330473813193708596, 14.460595500282343658512455029761, 15.24317477961960900534035876272, 16.28197482280994643671680895117, 17.07896671350634967237063916790, 17.85748153467536448086092807730, 18.502555812650532814630132115867, 19.58101811539811948676856925688, 19.83984059473122310498317020279, 20.76662176651096043029700095577

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