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Properties

Label 1-1183-1183.16-r0-0-0
Degree $1$
Conductor $1183$
Sign $0.673 - 0.739i$
Analytic cond. $5.49382$
Root an. cond. $5.49382$
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  + (−0.354 − 0.935i)2-s + (−0.996 − 0.0804i)3-s + (−0.748 + 0.663i)4-s + (0.799 − 0.600i)5-s + (0.278 + 0.960i)6-s + (0.885 + 0.464i)8-s + (0.987 + 0.160i)9-s + (−0.845 − 0.534i)10-s + (0.987 − 0.160i)11-s + (0.799 − 0.600i)12-s + (−0.845 + 0.534i)15-s + (0.120 − 0.992i)16-s + (0.885 + 0.464i)17-s + (−0.200 − 0.979i)18-s + (−0.5 + 0.866i)19-s + (−0.200 + 0.979i)20-s + ⋯
L(s)  = 1  + (−0.354 − 0.935i)2-s + (−0.996 − 0.0804i)3-s + (−0.748 + 0.663i)4-s + (0.799 − 0.600i)5-s + (0.278 + 0.960i)6-s + (0.885 + 0.464i)8-s + (0.987 + 0.160i)9-s + (−0.845 − 0.534i)10-s + (0.987 − 0.160i)11-s + (0.799 − 0.600i)12-s + (−0.845 + 0.534i)15-s + (0.120 − 0.992i)16-s + (0.885 + 0.464i)17-s + (−0.200 − 0.979i)18-s + (−0.5 + 0.866i)19-s + (−0.200 + 0.979i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.673 - 0.739i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.673 - 0.739i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1183\)    =    \(7 \cdot 13^{2}\)
Sign: $0.673 - 0.739i$
Analytic conductor: \(5.49382\)
Root analytic conductor: \(5.49382\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1183} (16, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1183,\ (0:\ ),\ 0.673 - 0.739i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.025970905 - 0.4532628207i\)
\(L(\frac12)\) \(\approx\) \(1.025970905 - 0.4532628207i\)
\(L(1)\) \(\approx\) \(0.7621971061 - 0.3381033785i\)
\(L(1)\) \(\approx\) \(0.7621971061 - 0.3381033785i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
13 \( 1 \)
good2 \( 1 + (-0.354 - 0.935i)T \)
3 \( 1 + (-0.996 - 0.0804i)T \)
5 \( 1 + (0.799 - 0.600i)T \)
11 \( 1 + (0.987 - 0.160i)T \)
17 \( 1 + (0.885 + 0.464i)T \)
19 \( 1 + (-0.5 + 0.866i)T \)
23 \( 1 + T \)
29 \( 1 + (0.987 + 0.160i)T \)
31 \( 1 + (0.692 - 0.721i)T \)
37 \( 1 + (-0.970 - 0.239i)T \)
41 \( 1 + (0.428 + 0.903i)T \)
43 \( 1 + (0.692 + 0.721i)T \)
47 \( 1 + (-0.200 + 0.979i)T \)
53 \( 1 + (-0.845 + 0.534i)T \)
59 \( 1 + (0.120 + 0.992i)T \)
61 \( 1 + (-0.0402 + 0.999i)T \)
67 \( 1 + (-0.200 + 0.979i)T \)
71 \( 1 + (-0.996 - 0.0804i)T \)
73 \( 1 + (-0.632 - 0.774i)T \)
79 \( 1 + (-0.200 + 0.979i)T \)
83 \( 1 + (0.568 + 0.822i)T \)
89 \( 1 + T \)
97 \( 1 + (-0.919 + 0.391i)T \)
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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

−21.673428409204746858980208907755, −20.74146257078634881477543199960, −19.23726270155756794005949433969, −18.936331726691329539100409762906, −17.86331892150247330480063380875, −17.387858783358747352494655545828, −16.97836090632089862402894924701, −15.993061269115264785642537362957, −15.290403082791256218811488090923, −14.4035295786022558393877647963, −13.78833697102417657730521921401, −12.82808785948969005326534268427, −11.8694527035828348696527059648, −10.84613560508457145618583869379, −10.22551463558146621027393997118, −9.46920897759789749241280162200, −8.72190886235445603682707770627, −7.31495709961645806664324896344, −6.76652416098453877041073076828, −6.20426370393882983058304900033, −5.25851621988275537522050405762, −4.6493029411547856775673119016, −3.37263500072258840563762024390, −1.78969381876515331366446770547, −0.77546954393103089297887129357, 1.12173816433962205538484530767, 1.40115844979759685562172914667, 2.75313514065227058462285013555, 4.03746976242878083522115110450, 4.70806650935005441154625476649, 5.72522672561935440904029858647, 6.42358353015873522345181585154, 7.65808854267730984770782438235, 8.63719606336299099987110628168, 9.473096335693966865439320631256, 10.157530754833536551299097286997, 10.83199769950300044053810720178, 11.80543163859698861693218769274, 12.37116270360298913925230735440, 12.979125548352006421069455565594, 13.86310565792883226170451881376, 14.72945317765195666523712949849, 16.27906149824806631804296515807, 16.70941275852354289632086150384, 17.444678795663775959735003122018, 17.830249906526794952786658401863, 19.03156944681966237261280141103, 19.28666871292826190020326376867, 20.59093035513227636583696389662, 21.181722861753674633938111319777

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