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Properties

Label 1-1083-1083.278-r0-0-0
Degree $1$
Conductor $1083$
Sign $0.998 + 0.0507i$
Analytic cond. $5.02943$
Root an. cond. $5.02943$
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  + (−0.962 − 0.272i)2-s + (0.851 + 0.523i)4-s + (−0.451 − 0.892i)5-s + (−0.879 − 0.475i)7-s + (−0.677 − 0.735i)8-s + (0.191 + 0.981i)10-s + (0.986 − 0.164i)11-s + (−0.904 + 0.426i)13-s + (0.716 + 0.697i)14-s + (0.451 + 0.892i)16-s + (−0.851 + 0.523i)17-s + (0.0825 − 0.996i)20-s + (−0.993 − 0.110i)22-s + (0.821 − 0.569i)23-s + (−0.592 + 0.805i)25-s + (0.986 − 0.164i)26-s + ⋯
L(s)  = 1  + (−0.962 − 0.272i)2-s + (0.851 + 0.523i)4-s + (−0.451 − 0.892i)5-s + (−0.879 − 0.475i)7-s + (−0.677 − 0.735i)8-s + (0.191 + 0.981i)10-s + (0.986 − 0.164i)11-s + (−0.904 + 0.426i)13-s + (0.716 + 0.697i)14-s + (0.451 + 0.892i)16-s + (−0.851 + 0.523i)17-s + (0.0825 − 0.996i)20-s + (−0.993 − 0.110i)22-s + (0.821 − 0.569i)23-s + (−0.592 + 0.805i)25-s + (0.986 − 0.164i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1083 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0507i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1083 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0507i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1083\)    =    \(3 \cdot 19^{2}\)
Sign: $0.998 + 0.0507i$
Analytic conductor: \(5.02943\)
Root analytic conductor: \(5.02943\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1083} (278, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1083,\ (0:\ ),\ 0.998 + 0.0507i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5896856660 + 0.01497074179i\)
\(L(\frac12)\) \(\approx\) \(0.5896856660 + 0.01497074179i\)
\(L(1)\) \(\approx\) \(0.5610135797 - 0.1100030711i\)
\(L(1)\) \(\approx\) \(0.5610135797 - 0.1100030711i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
19 \( 1 \)
good2 \( 1 + (-0.962 - 0.272i)T \)
5 \( 1 + (-0.451 - 0.892i)T \)
7 \( 1 + (-0.879 - 0.475i)T \)
11 \( 1 + (0.986 - 0.164i)T \)
13 \( 1 + (-0.904 + 0.426i)T \)
17 \( 1 + (-0.851 + 0.523i)T \)
23 \( 1 + (0.821 - 0.569i)T \)
29 \( 1 + (0.0275 + 0.999i)T \)
31 \( 1 + (-0.245 + 0.969i)T \)
37 \( 1 + (0.986 - 0.164i)T \)
41 \( 1 + (-0.926 + 0.376i)T \)
43 \( 1 + (-0.191 + 0.981i)T \)
47 \( 1 + (-0.635 - 0.771i)T \)
53 \( 1 + (0.635 + 0.771i)T \)
59 \( 1 + (-0.926 + 0.376i)T \)
61 \( 1 + (0.975 + 0.218i)T \)
67 \( 1 + (0.298 - 0.954i)T \)
71 \( 1 + (0.975 - 0.218i)T \)
73 \( 1 + (0.851 - 0.523i)T \)
79 \( 1 + (0.191 - 0.981i)T \)
83 \( 1 + (-0.546 - 0.837i)T \)
89 \( 1 + (0.851 + 0.523i)T \)
97 \( 1 + (0.298 + 0.954i)T \)
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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

−21.52054441074887048941446487852, −20.19347511666548987415882434488, −19.73403504701876476353263574683, −19.019477579699553543090673547977, −18.509096011074428737194446633566, −17.50231994334694550238940513557, −16.95383298016135443613305345605, −15.93407799419807475592725517600, −15.22322016790586256949211718050, −14.84794069932276993732746158348, −13.73863224246248005456624112772, −12.55406226747154453126019426139, −11.61089697800635428903699221961, −11.19397881658539730236059908852, −9.91409900807361172722372079013, −9.6246008858685959237908593011, −8.64346367811989669716192138572, −7.625199573044428626857561569664, −6.87145656123854733452154586632, −6.38087344481461569839195264434, −5.30189887021102581289526352155, −3.87642608355717667715625549219, −2.823248520244514439318689609236, −2.15095517900532981819170189819, −0.47403918659455893656338002829, 0.79844300423300718472409036195, 1.780577479072308095630516217717, 3.07777563588262417049106829513, 3.93173442263359571156803416262, 4.87793156665951469624483404346, 6.411386917531369717489639192660, 6.89588153814786061591225694893, 7.866923980710254879851810970375, 8.93941757693977930620785094576, 9.19179049149110109538505739901, 10.19089661163079429135913976692, 11.07511675438893603836522180660, 11.934771474536797622398931957631, 12.60101215602728649444054925355, 13.26477671101917639438144182740, 14.588037586722414140796493224995, 15.43824576834965282397288251640, 16.52556239743249763049217922796, 16.62951048891239924999972986457, 17.40047910626035624714395618950, 18.45060983709550772042804007641, 19.44345840550013665975713793807, 19.800634162630841376572259352516, 20.20126276510295626819559402572, 21.42529754448853251089483504092

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