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Properties

Label 1-1183-1183.482-r1-0-0
Degree $1$
Conductor $1183$
Sign $-0.102 - 0.994i$
Analytic cond. $127.131$
Root an. cond. $127.131$
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  + (−0.748 − 0.663i)2-s + (0.354 − 0.935i)3-s + (0.120 + 0.992i)4-s + (0.970 + 0.239i)5-s + (−0.885 + 0.464i)6-s + (0.568 − 0.822i)8-s + (−0.748 − 0.663i)9-s + (−0.568 − 0.822i)10-s + (−0.748 + 0.663i)11-s + (0.970 + 0.239i)12-s + (0.568 − 0.822i)15-s + (−0.970 + 0.239i)16-s + (−0.568 + 0.822i)17-s + (0.120 + 0.992i)18-s − 19-s + (−0.120 + 0.992i)20-s + ⋯
L(s)  = 1  + (−0.748 − 0.663i)2-s + (0.354 − 0.935i)3-s + (0.120 + 0.992i)4-s + (0.970 + 0.239i)5-s + (−0.885 + 0.464i)6-s + (0.568 − 0.822i)8-s + (−0.748 − 0.663i)9-s + (−0.568 − 0.822i)10-s + (−0.748 + 0.663i)11-s + (0.970 + 0.239i)12-s + (0.568 − 0.822i)15-s + (−0.970 + 0.239i)16-s + (−0.568 + 0.822i)17-s + (0.120 + 0.992i)18-s − 19-s + (−0.120 + 0.992i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.102 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.102 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1183\)    =    \(7 \cdot 13^{2}\)
Sign: $-0.102 - 0.994i$
Analytic conductor: \(127.131\)
Root analytic conductor: \(127.131\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1183} (482, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1183,\ (1:\ ),\ -0.102 - 0.994i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.047735719 - 1.160732360i\)
\(L(\frac12)\) \(\approx\) \(1.047735719 - 1.160732360i\)
\(L(1)\) \(\approx\) \(0.7981884082 - 0.4237639217i\)
\(L(1)\) \(\approx\) \(0.7981884082 - 0.4237639217i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
13 \( 1 \)
good2 \( 1 + (-0.748 - 0.663i)T \)
3 \( 1 + (0.354 - 0.935i)T \)
5 \( 1 + (0.970 + 0.239i)T \)
11 \( 1 + (-0.748 + 0.663i)T \)
17 \( 1 + (-0.568 + 0.822i)T \)
19 \( 1 - T \)
23 \( 1 + T \)
29 \( 1 + (-0.748 - 0.663i)T \)
31 \( 1 + (-0.885 + 0.464i)T \)
37 \( 1 + (0.885 - 0.464i)T \)
41 \( 1 + (0.354 - 0.935i)T \)
43 \( 1 + (0.885 + 0.464i)T \)
47 \( 1 + (-0.120 + 0.992i)T \)
53 \( 1 + (0.568 - 0.822i)T \)
59 \( 1 + (0.970 + 0.239i)T \)
61 \( 1 + (-0.568 - 0.822i)T \)
67 \( 1 + (0.120 - 0.992i)T \)
71 \( 1 + (-0.354 + 0.935i)T \)
73 \( 1 + (0.748 - 0.663i)T \)
79 \( 1 + (0.120 - 0.992i)T \)
83 \( 1 + (0.354 + 0.935i)T \)
89 \( 1 - T \)
97 \( 1 + (0.970 - 0.239i)T \)
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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

−21.12491358625062080412464190430, −20.476429472594850374594683941794, −19.75130528093738982319697674061, −18.72224825691751524858579973845, −18.15056837979756147356392520477, −17.12712988567831675385366059881, −16.63174617434724598475044918008, −16.00904993735702362016470359381, −15.06909773178453034709971891297, −14.551917228673553312501120994198, −13.578982556059346140281039103813, −13.06688945274962084508622893430, −11.29123949392294532913567672498, −10.753335119968109863388410589257, −10.02613592601771437071874133548, −9.12813659960611752203840973449, −8.83281616737653004884177649461, −7.844264039688543041065581199143, −6.795820492172683447006787100051, −5.7451020829247050845403750757, −5.24458626899188627404276313293, −4.33811218019859205419647831708, −2.83335101623280056132604523579, −2.10914555288804499319303160177, −0.69986571543479757498664376742, 0.5191539444868394078411700322, 1.788403254115621758006058649521, 2.18258507800354412978301674627, 3.042490885982689034323170898732, 4.24680249269295995029343689387, 5.63374582154986759219978706153, 6.59495737015012178942581546341, 7.29693990526367854991389321256, 8.13820682617387819971796490429, 9.01002386179399019662372572494, 9.59960472317276197315353683042, 10.701397759245945799418963401631, 11.11723103966736578019779795172, 12.52913750932283731650002680286, 12.847435450094118391911923713736, 13.446172769883136203500374698723, 14.541736211828787431716547185849, 15.2854803902424015565841760622, 16.60961742745392737950165079265, 17.43876980829070189927691984150, 17.76646305378528373865502423926, 18.56142017180804716963839372223, 19.197313097085211686200368792869, 19.90114994018508341482495465354, 20.88643777779017418590747893828

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