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Properties

Label 1-1183-1183.1021-r0-0-0
Degree $1$
Conductor $1183$
Sign $0.660 + 0.750i$
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  + (−0.391 − 0.919i)2-s + (−0.948 − 0.316i)3-s + (−0.692 + 0.721i)4-s + (−0.464 − 0.885i)5-s + (0.0804 + 0.996i)6-s + (0.935 + 0.354i)8-s + (0.799 + 0.600i)9-s + (−0.632 + 0.774i)10-s + (−0.600 − 0.799i)11-s + (0.885 − 0.464i)12-s + (0.160 + 0.987i)15-s + (−0.0402 − 0.999i)16-s + (−0.632 − 0.774i)17-s + (0.239 − 0.970i)18-s + (0.866 + 0.5i)19-s + (0.960 + 0.278i)20-s + ⋯
L(s)  = 1  + (−0.391 − 0.919i)2-s + (−0.948 − 0.316i)3-s + (−0.692 + 0.721i)4-s + (−0.464 − 0.885i)5-s + (0.0804 + 0.996i)6-s + (0.935 + 0.354i)8-s + (0.799 + 0.600i)9-s + (−0.632 + 0.774i)10-s + (−0.600 − 0.799i)11-s + (0.885 − 0.464i)12-s + (0.160 + 0.987i)15-s + (−0.0402 − 0.999i)16-s + (−0.632 − 0.774i)17-s + (0.239 − 0.970i)18-s + (0.866 + 0.5i)19-s + (0.960 + 0.278i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.660 + 0.750i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.660 + 0.750i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1183\)    =    \(7 \cdot 13^{2}\)
Sign: $0.660 + 0.750i$
Analytic conductor: \(5.49382\)
Root analytic conductor: \(5.49382\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1183} (1021, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1183,\ (0:\ ),\ 0.660 + 0.750i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.03785918181 + 0.01710961141i\)
\(L(\frac12)\) \(\approx\) \(0.03785918181 + 0.01710961141i\)
\(L(1)\) \(\approx\) \(0.3617629185 - 0.2887782867i\)
\(L(1)\) \(\approx\) \(0.3617629185 - 0.2887782867i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
13 \( 1 \)
good2 \( 1 + (-0.391 - 0.919i)T \)
3 \( 1 + (-0.948 - 0.316i)T \)
5 \( 1 + (-0.464 - 0.885i)T \)
11 \( 1 + (-0.600 - 0.799i)T \)
17 \( 1 + (-0.632 - 0.774i)T \)
19 \( 1 + (0.866 + 0.5i)T \)
23 \( 1 + (0.5 + 0.866i)T \)
29 \( 1 + (-0.919 + 0.391i)T \)
31 \( 1 + (0.822 - 0.568i)T \)
37 \( 1 + (-0.903 - 0.428i)T \)
41 \( 1 + (0.316 - 0.948i)T \)
43 \( 1 + (-0.428 - 0.903i)T \)
47 \( 1 + (-0.239 - 0.970i)T \)
53 \( 1 + (-0.354 + 0.935i)T \)
59 \( 1 + (-0.999 - 0.0402i)T \)
61 \( 1 + (-0.987 - 0.160i)T \)
67 \( 1 + (-0.721 + 0.692i)T \)
71 \( 1 + (0.979 - 0.200i)T \)
73 \( 1 + (-0.992 + 0.120i)T \)
79 \( 1 + (-0.970 + 0.239i)T \)
83 \( 1 + (-0.663 + 0.748i)T \)
89 \( 1 + (0.866 - 0.5i)T \)
97 \( 1 + (0.534 + 0.845i)T \)
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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

−21.35796146356480108058347880898, −20.22965391269087176727924194965, −19.28261903787054904354009866835, −18.51082252159801194033606235407, −17.89087427597022192944049787542, −17.37989055603883436800045456167, −16.43352678245674542712411303562, −15.65000592630480467216596095641, −15.2208760094425645643560425761, −14.521552114883465075737562405896, −13.3748659492604420712817185648, −12.55387612287081040389898223989, −11.45554762568390546378160099145, −10.730598987250555212851202710861, −10.11823572172320433114543547776, −9.32886719866652993021739631785, −8.13512041107592237726957602306, −7.328803407653907843543603390080, −6.64828567999025543656487245266, −6.00604812643706819783449129582, −4.854761765095673926753100676054, −4.389519168149860856022602313864, −3.08681067042785882568693952798, −1.55477304116126580247710090795, −0.02864332283743416401208733829, 0.935622268886139459061292421575, 1.864164555590221538884108931344, 3.17774203404776003691972936030, 4.14405492365311909876910822890, 5.11789398593646641515152028464, 5.61770262053741552953608910293, 7.19312535750652106598719103785, 7.77035637534272747073340223297, 8.75345464237756655600097452757, 9.50113220558888907867270937929, 10.51682775069294414364960559415, 11.25243921068246164176719946845, 11.841470258634150853992128260592, 12.469120744137337218679914777888, 13.401033111822606633322959456342, 13.72530085941618139415510817873, 15.55416803865017748723798252294, 16.12903716718149445131024778955, 16.913865301913862792861550856657, 17.47415272375023981781709379383, 18.476489837053781323885332639474, 18.85747260054625539724783907575, 19.78785780577107236137733886750, 20.56378745616065233153051129044, 21.21875284920789256280814003967

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