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Properties

Label 1-1183-1183.1126-r0-0-0
Degree $1$
Conductor $1183$
Sign $0.778 + 0.627i$
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  + (−0.239 − 0.970i)2-s + (−0.120 − 0.992i)3-s + (−0.885 + 0.464i)4-s + (0.822 − 0.568i)5-s + (−0.935 + 0.354i)6-s + (0.663 + 0.748i)8-s + (−0.970 + 0.239i)9-s + (−0.748 − 0.663i)10-s + (−0.239 + 0.970i)11-s + (0.568 + 0.822i)12-s + (−0.663 − 0.748i)15-s + (0.568 − 0.822i)16-s + (−0.748 + 0.663i)17-s + (0.464 + 0.885i)18-s − i·19-s + (−0.464 + 0.885i)20-s + ⋯
L(s)  = 1  + (−0.239 − 0.970i)2-s + (−0.120 − 0.992i)3-s + (−0.885 + 0.464i)4-s + (0.822 − 0.568i)5-s + (−0.935 + 0.354i)6-s + (0.663 + 0.748i)8-s + (−0.970 + 0.239i)9-s + (−0.748 − 0.663i)10-s + (−0.239 + 0.970i)11-s + (0.568 + 0.822i)12-s + (−0.663 − 0.748i)15-s + (0.568 − 0.822i)16-s + (−0.748 + 0.663i)17-s + (0.464 + 0.885i)18-s − i·19-s + (−0.464 + 0.885i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.778 + 0.627i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.778 + 0.627i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1183\)    =    \(7 \cdot 13^{2}\)
Sign: $0.778 + 0.627i$
Analytic conductor: \(5.49382\)
Root analytic conductor: \(5.49382\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1183} (1126, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1183,\ (0:\ ),\ 0.778 + 0.627i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2035397812 + 0.07182892085i\)
\(L(\frac12)\) \(\approx\) \(0.2035397812 + 0.07182892085i\)
\(L(1)\) \(\approx\) \(0.5362362863 - 0.4506338432i\)
\(L(1)\) \(\approx\) \(0.5362362863 - 0.4506338432i\)

Euler product

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

−21.33150262351192136702746100300, −20.53835047075788565296030914769, −19.45850447561417669979682076503, −18.51752291577367852458961308979, −17.95783493018247777002572442134, −17.145517253205366795608328712757, −16.388271020409665723567745239417, −15.91057635688709755637048852153, −14.97082839626904698806046710384, −14.265468987312314845825935521479, −13.80848163296104972477931498368, −12.828185385703361467012683271159, −11.33801893907597560622844583037, −10.724766827701297172713977403775, −9.851306124445182227634392293981, −9.33686112826742561382935624953, −8.47722808481409289548148508962, −7.5605514713182574382869420671, −6.397052107940629715677440059789, −5.817177674406157527535140152967, −5.19327285828041649549711117194, −4.06308117953780227580680843500, −3.23517521665189941139087016305, −1.90170577052840077664845923380, −0.092387831978063730540695324201, 1.35913388504330987190837935766, 1.987856236877785480478737127693, 2.6872797366151535235994014631, 4.09870902409449461091114412628, 5.03716919104526734948183948920, 5.86377956816693696119142658638, 6.98574824201459839596632819346, 7.83760824224850314840122274098, 8.830346200869265474636575948360, 9.31746071510172881591478529697, 10.398295748943851194622404323, 11.080193267289636466100107378529, 12.15061916212926279561055051401, 12.65306188451597197118155621710, 13.265514349850470709163862034000, 13.898416579436933200184040856189, 14.82307489183916290927620575981, 16.17469148952096717407079747888, 17.22440679469062496924335761487, 17.64482050181825934681121594746, 18.11241830333851197686172567344, 19.05046226012002917403403292919, 19.90870684792624150775965723246, 20.31042631379259412098540171954, 21.132701307433996227814278119933

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