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Properties

Label 1-1183-1183.1175-r0-0-0
Degree $1$
Conductor $1183$
Sign $-0.319 + 0.947i$
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  + (−0.992 − 0.120i)2-s + (0.748 − 0.663i)3-s + (0.970 + 0.239i)4-s + (0.464 + 0.885i)5-s + (−0.822 + 0.568i)6-s + (−0.935 − 0.354i)8-s + (0.120 − 0.992i)9-s + (−0.354 − 0.935i)10-s + (−0.992 + 0.120i)11-s + (0.885 − 0.464i)12-s + (0.935 + 0.354i)15-s + (0.885 + 0.464i)16-s + (−0.354 + 0.935i)17-s + (−0.239 + 0.970i)18-s + i·19-s + (0.239 + 0.970i)20-s + ⋯
L(s)  = 1  + (−0.992 − 0.120i)2-s + (0.748 − 0.663i)3-s + (0.970 + 0.239i)4-s + (0.464 + 0.885i)5-s + (−0.822 + 0.568i)6-s + (−0.935 − 0.354i)8-s + (0.120 − 0.992i)9-s + (−0.354 − 0.935i)10-s + (−0.992 + 0.120i)11-s + (0.885 − 0.464i)12-s + (0.935 + 0.354i)15-s + (0.885 + 0.464i)16-s + (−0.354 + 0.935i)17-s + (−0.239 + 0.970i)18-s + i·19-s + (0.239 + 0.970i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.319 + 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.319 + 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1183\)    =    \(7 \cdot 13^{2}\)
Sign: $-0.319 + 0.947i$
Analytic conductor: \(5.49382\)
Root analytic conductor: \(5.49382\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1183} (1175, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1183,\ (0:\ ),\ -0.319 + 0.947i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3699558847 + 0.5152185324i\)
\(L(\frac12)\) \(\approx\) \(0.3699558847 + 0.5152185324i\)
\(L(1)\) \(\approx\) \(0.7515895712 + 0.03793168010i\)
\(L(1)\) \(\approx\) \(0.7515895712 + 0.03793168010i\)

Euler product

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

−20.781837310684465809643501371910, −20.07804339355758444933867361199, −19.78790761264126046487204618238, −18.519960687692672677502738860288, −18.04482287327746585433052367819, −17.023366111514522841933371474077, −16.291069722941138939784925976535, −15.80118059989742892194523042683, −15.110662685608932408994896081950, −13.99449833949294626798395918980, −13.35014913645501620369436427715, −12.35105287588721854049489227787, −11.25191440913972715323598449238, −10.465931544170742953052674539656, −9.71395484785633355948349119517, −9.08934816438055863385085624763, −8.44141460710115051392992946267, −7.712875610084692763109567792749, −6.74543334538927421278828888326, −5.38614080904576639963678739675, −4.958006736836866386593688471058, −3.55216231386730880910343202329, −2.47369434026345919978093872945, −1.83405815736306484779427908325, −0.28307378265212328995179406809, 1.551093843756094651537575217470, 2.14874589624421385123818087480, 2.99381060571084526073331019268, 3.8408724910374619923882842440, 5.748960601426002668923052587456, 6.39623137366889432269687335397, 7.26541056437945549327199634808, 7.96787258926122895347710840811, 8.56814461946685728334224866223, 9.71556512483200725823986638325, 10.207793177174674364177484678352, 11.00924156099448002409166703860, 12.075512216994357920570484296270, 12.79831009546535903617040956526, 13.72186135159247456393234291565, 14.54897989742448290114450073561, 15.284882661398333246448850569279, 15.97991600055253037750327993190, 17.27168593558939123183858956535, 17.73732753657626498960288321046, 18.67140153934384423896528791353, 18.79068267274476193840966382548, 19.77507063843522155784545733161, 20.52041945725957158246852667090, 21.19890354061479552452763305000

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