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Properties

Label 1-1183-1183.870-r0-0-0
Degree $1$
Conductor $1183$
Sign $-0.533 - 0.845i$
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  + (0.200 − 0.979i)2-s + (−0.632 − 0.774i)3-s + (−0.919 − 0.391i)4-s + (−0.278 − 0.960i)5-s + (−0.885 + 0.464i)6-s + (−0.568 + 0.822i)8-s + (−0.200 + 0.979i)9-s + (−0.996 + 0.0804i)10-s + (0.200 + 0.979i)11-s + (0.278 + 0.960i)12-s + (−0.568 + 0.822i)15-s + (0.692 + 0.721i)16-s + (0.428 + 0.903i)17-s + (0.919 + 0.391i)18-s + (0.5 + 0.866i)19-s + (−0.120 + 0.992i)20-s + ⋯
L(s)  = 1  + (0.200 − 0.979i)2-s + (−0.632 − 0.774i)3-s + (−0.919 − 0.391i)4-s + (−0.278 − 0.960i)5-s + (−0.885 + 0.464i)6-s + (−0.568 + 0.822i)8-s + (−0.200 + 0.979i)9-s + (−0.996 + 0.0804i)10-s + (0.200 + 0.979i)11-s + (0.278 + 0.960i)12-s + (−0.568 + 0.822i)15-s + (0.692 + 0.721i)16-s + (0.428 + 0.903i)17-s + (0.919 + 0.391i)18-s + (0.5 + 0.866i)19-s + (−0.120 + 0.992i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.533 - 0.845i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.533 - 0.845i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1183\)    =    \(7 \cdot 13^{2}\)
Sign: $-0.533 - 0.845i$
Analytic conductor: \(5.49382\)
Root analytic conductor: \(5.49382\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1183} (870, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1183,\ (0:\ ),\ -0.533 - 0.845i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4927516177 - 0.8931126785i\)
\(L(\frac12)\) \(\approx\) \(0.4927516177 - 0.8931126785i\)
\(L(1)\) \(\approx\) \(0.5932519907 - 0.5894198641i\)
\(L(1)\) \(\approx\) \(0.5932519907 - 0.5894198641i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
13 \( 1 \)
good2 \( 1 + (0.200 - 0.979i)T \)
3 \( 1 + (-0.632 - 0.774i)T \)
5 \( 1 + (-0.278 - 0.960i)T \)
11 \( 1 + (0.200 + 0.979i)T \)
17 \( 1 + (0.428 + 0.903i)T \)
19 \( 1 + (0.5 + 0.866i)T \)
23 \( 1 + (-0.5 - 0.866i)T \)
29 \( 1 + (-0.748 - 0.663i)T \)
31 \( 1 + (0.0402 - 0.999i)T \)
37 \( 1 + (0.845 + 0.534i)T \)
41 \( 1 + (0.354 - 0.935i)T \)
43 \( 1 + (0.885 + 0.464i)T \)
47 \( 1 + (0.919 - 0.391i)T \)
53 \( 1 + (0.428 + 0.903i)T \)
59 \( 1 + (-0.692 + 0.721i)T \)
61 \( 1 + (0.428 - 0.903i)T \)
67 \( 1 + (-0.799 - 0.600i)T \)
71 \( 1 + (0.354 - 0.935i)T \)
73 \( 1 + (0.200 + 0.979i)T \)
79 \( 1 + (-0.919 + 0.391i)T \)
83 \( 1 + (0.354 + 0.935i)T \)
89 \( 1 + (0.5 + 0.866i)T \)
97 \( 1 + (0.970 - 0.239i)T \)
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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

−21.90514486506673790465299460272, −21.11364741432303981881722603880, −19.898728130154478337106133544903, −18.91776596618467579501755933344, −18.09521653894082253433087251407, −17.64244682462538746362181226384, −16.600283539681782449684637590002, −15.99450498441003654638448791285, −15.52709205949718330835102089679, −14.50600197988347526974995382091, −14.12786917476415035811082932075, −13.10406589423051466273773955829, −11.87388026107233486448067565731, −11.35059034860979037876697542958, −10.43270382575786221105353075130, −9.49588480015932465092688848782, −8.86395795791495695716783659421, −7.62370171909086926696329801970, −7.00563285655380000050434310146, −6.038499484753398187013191682490, −5.49093323739949447534573218933, −4.497330804446492799364705971484, −3.539848950490715904391699075047, −2.98248851835099581709711244483, −0.71069803830146650862485454641, 0.72681590847713802476080352445, 1.62838218467783430637689356905, 2.368565500921092311270138800244, 3.92902431544297175149845082179, 4.48252312120816262399033894506, 5.543375556466295112749687833797, 6.10249219444061900420624074530, 7.59562649400346832471442298150, 8.12840122361659014498931374031, 9.22086632802021704381391754845, 10.02904661005224442672072213473, 10.888100658582367728584256328657, 11.87486226204197414411709124052, 12.29747827155743273986248861404, 12.82586455281130429317297830144, 13.61707084775943126391139568117, 14.52183930991310771127204567222, 15.49384144558993080242874170946, 16.80770842333311409392301007149, 17.06921348418234882046726962808, 18.079479848436585215258795798274, 18.73622527276170678278644293034, 19.468994642698794275387036753252, 20.26343696191923591425781009005, 20.728052786092459179190856568443

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