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Properties

Label 1-1183-1183.479-r0-0-0
Degree $1$
Conductor $1183$
Sign $0.999 + 0.0245i$
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  + (−0.979 − 0.200i)2-s + (0.354 − 0.935i)3-s + (0.919 + 0.391i)4-s + (−0.721 − 0.692i)5-s + (−0.534 + 0.845i)6-s + (−0.822 − 0.568i)8-s + (−0.748 − 0.663i)9-s + (0.568 + 0.822i)10-s + (0.663 + 0.748i)11-s + (0.692 − 0.721i)12-s + (−0.903 + 0.428i)15-s + (0.692 + 0.721i)16-s + (0.428 + 0.903i)17-s + (0.600 + 0.799i)18-s − i·19-s + (−0.391 − 0.919i)20-s + ⋯
L(s)  = 1  + (−0.979 − 0.200i)2-s + (0.354 − 0.935i)3-s + (0.919 + 0.391i)4-s + (−0.721 − 0.692i)5-s + (−0.534 + 0.845i)6-s + (−0.822 − 0.568i)8-s + (−0.748 − 0.663i)9-s + (0.568 + 0.822i)10-s + (0.663 + 0.748i)11-s + (0.692 − 0.721i)12-s + (−0.903 + 0.428i)15-s + (0.692 + 0.721i)16-s + (0.428 + 0.903i)17-s + (0.600 + 0.799i)18-s − i·19-s + (−0.391 − 0.919i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0245i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0245i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1183\)    =    \(7 \cdot 13^{2}\)
Sign: $0.999 + 0.0245i$
Analytic conductor: \(5.49382\)
Root analytic conductor: \(5.49382\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1183} (479, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1183,\ (0:\ ),\ 0.999 + 0.0245i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7592578892 + 0.009304875569i\)
\(L(\frac12)\) \(\approx\) \(0.7592578892 + 0.009304875569i\)
\(L(1)\) \(\approx\) \(0.6532988622 - 0.2053792109i\)
\(L(1)\) \(\approx\) \(0.6532988622 - 0.2053792109i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
13 \( 1 \)
good2 \( 1 + (-0.979 - 0.200i)T \)
3 \( 1 + (0.354 - 0.935i)T \)
5 \( 1 + (-0.721 - 0.692i)T \)
11 \( 1 + (0.663 + 0.748i)T \)
17 \( 1 + (0.428 + 0.903i)T \)
19 \( 1 - iT \)
23 \( 1 + (0.5 + 0.866i)T \)
29 \( 1 + (0.948 - 0.316i)T \)
31 \( 1 + (-0.534 + 0.845i)T \)
37 \( 1 + (-0.534 + 0.845i)T \)
41 \( 1 + (-0.774 + 0.632i)T \)
43 \( 1 + (0.0402 + 0.999i)T \)
47 \( 1 + (-0.600 + 0.799i)T \)
53 \( 1 + (-0.996 - 0.0804i)T \)
59 \( 1 + (0.721 + 0.692i)T \)
61 \( 1 + (-0.568 - 0.822i)T \)
67 \( 1 + (0.992 + 0.120i)T \)
71 \( 1 + (0.160 + 0.987i)T \)
73 \( 1 + (-0.316 + 0.948i)T \)
79 \( 1 + (0.799 + 0.600i)T \)
83 \( 1 + (-0.935 + 0.354i)T \)
89 \( 1 + (0.866 - 0.5i)T \)
97 \( 1 + (0.721 - 0.692i)T \)
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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

−20.94334043928781698452627547359, −20.38519576657671758882579781926, −19.551715704240243103962592102225, −18.94397051114958859718895786180, −18.35863669740430590390513008021, −17.19621453967393124046452530684, −16.42294206157435538984144848868, −16.03098778790005098608597894783, −15.1296665965026147880719586642, −14.48210345016899164518952005771, −13.93845623040427911680876372334, −12.173148539713468571373575571792, −11.54198893034176320313326505496, −10.730475454534915232326280478388, −10.22490621825911527808493435486, −9.25331800677265890409888171991, −8.56225119756553154021535745361, −7.8455899765484690538625151924, −6.94983706540097133118668364044, −6.04280180289269017247390744329, −5.01349886520468476504323690976, −3.71181454725596470264055624073, −3.15341237630938770658083227823, −2.09492581513868977029243617492, −0.46492257922532917123285101334, 1.10387216914985935677197245514, 1.60122903053621251529873340131, 2.88382269499107380066963614006, 3.71229812100436278168875821920, 4.9807529901943865859622915213, 6.353896088966803682219970765737, 7.0060167800149145135735646547, 7.804914607863198657609135670191, 8.45626413865107373434711285326, 9.13404598607230843148334398243, 9.94041311557719258482760445358, 11.258088503102369671198583498356, 11.74129220724672977786995152624, 12.59362644302661838603442748010, 13.01755342473832176985031564393, 14.33833969685022383234946778843, 15.209911585531620015143867488154, 15.84035408142778470866360332781, 17.05208821532502705317971459396, 17.32060186353147921210097908488, 18.18029623827862218991263255323, 19.15656162151924132134259181134, 19.62321696800863809496285111472, 20.031398934697881425830755790490, 20.88671469421498234039902452093

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