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Properties

Label 1-1183-1183.124-r0-0-0
Degree $1$
Conductor $1183$
Sign $0.560 - 0.828i$
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 + (0.748 + 0.663i)3-s + (−0.692 − 0.721i)4-s + (−0.999 + 0.0402i)5-s + (0.903 − 0.428i)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 + (−0.0402 − 0.999i)12-s + (−0.774 − 0.632i)15-s + (−0.0402 + 0.999i)16-s + (−0.632 + 0.774i)17-s + (0.960 + 0.278i)18-s − i·19-s + (0.721 + 0.692i)20-s + ⋯
L(s)  = 1  + (0.391 − 0.919i)2-s + (0.748 + 0.663i)3-s + (−0.692 − 0.721i)4-s + (−0.999 + 0.0402i)5-s + (0.903 − 0.428i)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 + (−0.0402 − 0.999i)12-s + (−0.774 − 0.632i)15-s + (−0.0402 + 0.999i)16-s + (−0.632 + 0.774i)17-s + (0.960 + 0.278i)18-s − i·19-s + (0.721 + 0.692i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.560 - 0.828i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.560 - 0.828i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1183\)    =    \(7 \cdot 13^{2}\)
Sign: $0.560 - 0.828i$
Analytic conductor: \(5.49382\)
Root analytic conductor: \(5.49382\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1183} (124, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1183,\ (0:\ ),\ 0.560 - 0.828i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.396118326 - 0.7407877721i\)
\(L(\frac12)\) \(\approx\) \(1.396118326 - 0.7407877721i\)
\(L(1)\) \(\approx\) \(1.144785624 - 0.3758559129i\)
\(L(1)\) \(\approx\) \(1.144785624 - 0.3758559129i\)

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 \( 1 + (0.748 + 0.663i)T \)
5 \( 1 + (-0.999 + 0.0402i)T \)
11 \( 1 + (-0.992 - 0.120i)T \)
17 \( 1 + (-0.632 + 0.774i)T \)
19 \( 1 - iT \)
23 \( 1 + (0.5 - 0.866i)T \)
29 \( 1 + (0.799 - 0.600i)T \)
31 \( 1 + (0.903 - 0.428i)T \)
37 \( 1 + (0.903 - 0.428i)T \)
41 \( 1 + (0.979 + 0.200i)T \)
43 \( 1 + (0.996 - 0.0804i)T \)
47 \( 1 + (-0.960 + 0.278i)T \)
53 \( 1 + (0.987 + 0.160i)T \)
59 \( 1 + (0.999 - 0.0402i)T \)
61 \( 1 + (0.354 - 0.935i)T \)
67 \( 1 + (0.239 - 0.970i)T \)
71 \( 1 + (0.316 + 0.948i)T \)
73 \( 1 + (-0.600 + 0.799i)T \)
79 \( 1 + (0.278 + 0.960i)T \)
83 \( 1 + (0.663 + 0.748i)T \)
89 \( 1 + (-0.866 - 0.5i)T \)
97 \( 1 + (0.999 + 0.0402i)T \)
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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

−21.15585673881815813179630840544, −20.71181098029755519571003972027, −19.66496862610796385054244051877, −19.009851738060095401848714857010, −18.142142509102001736715991565323, −17.69936079684574243388999257307, −16.35858232192702214405044522731, −15.85096878196214697880973049966, −15.11023221322042581435383587442, −14.47545776978376361223646847273, −13.5809337888047946560767160653, −12.96641256548132739009618826372, −12.21378122878364136008798297731, −11.484060623395542686861031635316, −10.1038446691806670817269993150, −9.0123707606747517688506086202, −8.3503745276608571098942930140, −7.62613770892939691616249062646, −7.16626950042707384719849797999, −6.20915285465605609586103944303, −5.08436598807083918839253899119, −4.22576665753760657108079475291, −3.28311742073779742184419936395, −2.57067173542057989153913143383, −0.854596861023805702456113011923, 0.71723236909433690664184611786, 2.47706753270604328455764072157, 2.749451864798247287493311091089, 3.96967801217410004614065666936, 4.436624522496824055985459205582, 5.248441123618127150568100964229, 6.569333085304435412531259086837, 7.899379194133090487139645698019, 8.45258552484872576468354994409, 9.286411344159538238515247287872, 10.25153549305196616975792561457, 10.91519442167582741789848952898, 11.452767673440306749123562131808, 12.704340251893200281339294558830, 13.135437774873828370262824197002, 14.10589275918254804077414894663, 14.921085686387769918313297527918, 15.490741430532881432203123716779, 16.09598799904889142674801963492, 17.36865293728050870279553844806, 18.39124729960167310149176670701, 19.19652576839249152638791712800, 19.64549514232458677653150897389, 20.32205599063931338133803116811, 21.11044172642509634082302449493

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