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Properties

Label 1-1183-1183.18-r1-0-0
Degree $1$
Conductor $1183$
Sign $-0.946 + 0.323i$
Analytic cond. $127.131$
Root an. cond. $127.131$
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  + (0.903 − 0.428i)2-s + (−0.845 − 0.534i)3-s + (0.632 − 0.774i)4-s + (−0.316 − 0.948i)5-s + (−0.992 − 0.120i)6-s + (0.239 − 0.970i)8-s + (0.428 + 0.903i)9-s + (−0.692 − 0.721i)10-s + (0.903 + 0.428i)11-s + (−0.948 + 0.316i)12-s + (−0.239 + 0.970i)15-s + (−0.200 − 0.979i)16-s + (−0.278 − 0.960i)17-s + (0.774 + 0.632i)18-s + (0.866 + 0.5i)19-s + (−0.935 − 0.354i)20-s + ⋯
L(s)  = 1  + (0.903 − 0.428i)2-s + (−0.845 − 0.534i)3-s + (0.632 − 0.774i)4-s + (−0.316 − 0.948i)5-s + (−0.992 − 0.120i)6-s + (0.239 − 0.970i)8-s + (0.428 + 0.903i)9-s + (−0.692 − 0.721i)10-s + (0.903 + 0.428i)11-s + (−0.948 + 0.316i)12-s + (−0.239 + 0.970i)15-s + (−0.200 − 0.979i)16-s + (−0.278 − 0.960i)17-s + (0.774 + 0.632i)18-s + (0.866 + 0.5i)19-s + (−0.935 − 0.354i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.946 + 0.323i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.946 + 0.323i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1183\)    =    \(7 \cdot 13^{2}\)
Sign: $-0.946 + 0.323i$
Analytic conductor: \(127.131\)
Root analytic conductor: \(127.131\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1183} (18, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1183,\ (1:\ ),\ -0.946 + 0.323i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.4066235316 - 2.443322809i\)
\(L(\frac12)\) \(\approx\) \(-0.4066235316 - 2.443322809i\)
\(L(1)\) \(\approx\) \(0.9976080981 - 0.9887348854i\)
\(L(1)\) \(\approx\) \(0.9976080981 - 0.9887348854i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
13 \( 1 \)
good2 \( 1 + (0.903 - 0.428i)T \)
3 \( 1 + (-0.845 - 0.534i)T \)
5 \( 1 + (-0.316 - 0.948i)T \)
11 \( 1 + (0.903 + 0.428i)T \)
17 \( 1 + (-0.278 - 0.960i)T \)
19 \( 1 + (0.866 + 0.5i)T \)
23 \( 1 + (0.5 - 0.866i)T \)
29 \( 1 + (0.568 - 0.822i)T \)
31 \( 1 + (0.391 + 0.919i)T \)
37 \( 1 + (-0.600 + 0.799i)T \)
41 \( 1 + (-0.464 - 0.885i)T \)
43 \( 1 + (-0.120 - 0.992i)T \)
47 \( 1 + (0.774 - 0.632i)T \)
53 \( 1 + (0.278 + 0.960i)T \)
59 \( 1 + (-0.979 - 0.200i)T \)
61 \( 1 + (0.278 - 0.960i)T \)
67 \( 1 + (-0.160 - 0.987i)T \)
71 \( 1 + (-0.464 - 0.885i)T \)
73 \( 1 + (0.903 + 0.428i)T \)
79 \( 1 + (-0.632 - 0.774i)T \)
83 \( 1 + (0.464 - 0.885i)T \)
89 \( 1 + (-0.866 - 0.5i)T \)
97 \( 1 + (0.663 + 0.748i)T \)
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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

−21.72903441043499103811104264596, −21.14085527281417301854757673560, −19.96681273969330340668532937112, −19.29820037902914540325811337371, −18.06121536396708940757997989236, −17.46274570046833226544110214966, −16.68924479474292815631238572954, −15.897334182021884115092221410816, −15.252719303195156179521543747820, −14.63501442031428530322367248211, −13.83625193065373555010158842100, −12.86297297301322938036912506631, −11.89560276212121976762702243225, −11.36386995392434481664496837658, −10.811776867888889481549550274659, −9.760865296185126817123097314592, −8.66395349574895529028739331037, −7.4855337072026039787889339781, −6.74279715471365005533550147157, −6.13056855718726260628081518643, −5.32863465181209074620354386189, −4.2688039495568872312451110592, −3.640944475561086611395036827572, −2.831626959198905833059820918836, −1.28675294843969266913895912369, 0.4356136820468969425759078293, 1.20343195563088386013178675867, 2.0914597609126195446925581424, 3.41743346401551954543319719852, 4.5569830203489480033053094851, 4.93875048224699560991442066581, 5.87932513821536003018431190292, 6.77895295536696913470199218024, 7.44103574222022320364454542559, 8.70269960688584430253881117469, 9.72231707602203550155539687932, 10.59140881724028453880120074145, 11.59309314573938109459840313154, 12.161053001834389059526567429365, 12.41284317275967527434174939672, 13.65318150640322912570793748324, 13.9138500609552662988916837240, 15.297502293834200167906179319822, 15.918288026364201398078579939188, 16.72849234248445269183214863877, 17.362748881992128882225007854237, 18.4978764735676386497316762375, 19.14180338157206778065015257448, 20.0937968046959252568439986143, 20.496574189764499171393697956697

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