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Properties

Label 1-1083-1083.8-r0-0-0
Degree $1$
Conductor $1083$
Sign $0.977 - 0.208i$
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  + (−0.998 − 0.0550i)2-s + (0.993 + 0.110i)4-s + (−0.975 − 0.218i)5-s + (−0.401 + 0.915i)7-s + (−0.986 − 0.164i)8-s + (0.962 + 0.272i)10-s + (−0.789 + 0.614i)11-s + (0.754 − 0.656i)13-s + (0.451 − 0.892i)14-s + (0.975 + 0.218i)16-s + (−0.993 + 0.110i)17-s + (−0.945 − 0.324i)20-s + (0.821 − 0.569i)22-s + (0.191 − 0.981i)23-s + (0.904 + 0.426i)25-s + (−0.789 + 0.614i)26-s + ⋯
L(s)  = 1  + (−0.998 − 0.0550i)2-s + (0.993 + 0.110i)4-s + (−0.975 − 0.218i)5-s + (−0.401 + 0.915i)7-s + (−0.986 − 0.164i)8-s + (0.962 + 0.272i)10-s + (−0.789 + 0.614i)11-s + (0.754 − 0.656i)13-s + (0.451 − 0.892i)14-s + (0.975 + 0.218i)16-s + (−0.993 + 0.110i)17-s + (−0.945 − 0.324i)20-s + (0.821 − 0.569i)22-s + (0.191 − 0.981i)23-s + (0.904 + 0.426i)25-s + (−0.789 + 0.614i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1083 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.977 - 0.208i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1083 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.977 - 0.208i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1083\)    =    \(3 \cdot 19^{2}\)
Sign: $0.977 - 0.208i$
Analytic conductor: \(5.02943\)
Root analytic conductor: \(5.02943\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1083} (8, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1083,\ (0:\ ),\ 0.977 - 0.208i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5047516647 - 0.05327360349i\)
\(L(\frac12)\) \(\approx\) \(0.5047516647 - 0.05327360349i\)
\(L(1)\) \(\approx\) \(0.5141519600 + 0.005990986381i\)
\(L(1)\) \(\approx\) \(0.5141519600 + 0.005990986381i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
19 \( 1 \)
good2 \( 1 + (-0.998 - 0.0550i)T \)
5 \( 1 + (-0.975 - 0.218i)T \)
7 \( 1 + (-0.401 + 0.915i)T \)
11 \( 1 + (-0.789 + 0.614i)T \)
13 \( 1 + (0.754 - 0.656i)T \)
17 \( 1 + (-0.993 + 0.110i)T \)
23 \( 1 + (0.191 - 0.981i)T \)
29 \( 1 + (-0.592 - 0.805i)T \)
31 \( 1 + (-0.546 - 0.837i)T \)
37 \( 1 + (-0.789 + 0.614i)T \)
41 \( 1 + (0.851 + 0.523i)T \)
43 \( 1 + (-0.962 + 0.272i)T \)
47 \( 1 + (-0.137 - 0.990i)T \)
53 \( 1 + (0.137 + 0.990i)T \)
59 \( 1 + (0.851 + 0.523i)T \)
61 \( 1 + (0.350 - 0.936i)T \)
67 \( 1 + (-0.635 + 0.771i)T \)
71 \( 1 + (0.350 + 0.936i)T \)
73 \( 1 + (0.993 - 0.110i)T \)
79 \( 1 + (0.962 - 0.272i)T \)
83 \( 1 + (0.677 + 0.735i)T \)
89 \( 1 + (0.993 + 0.110i)T \)
97 \( 1 + (-0.635 - 0.771i)T \)
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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

−21.15498274132200686949940324269, −20.53102512986328333780999124174, −19.57864530461996826398153788245, −19.33856389670640419155090371793, −18.36638937624812479467483324016, −17.75202687268964238392934775455, −16.65856343507906312110063967894, −16.05501055557992459250993068070, −15.67122826328154435805561297160, −14.594917350777726530017719539521, −13.60728319198333991636775055808, −12.7229351511108339567076184754, −11.611183524830432021682473929, −10.90441537106560816022884494346, −10.590140493737551683081997524263, −9.31042278473710000522822387853, −8.64304637133578201240979283360, −7.73214562297696600268162246075, −7.08374864354054065954034747208, −6.40471920518515813526957016012, −5.140131782254704780612116520841, −3.76374742029167365597773411172, −3.24351270181863638901717531127, −1.88351392525419275152507178945, −0.63003583211283114138927196003, 0.50565489792548201489341466826, 2.03105549725317014073993364557, 2.83440142930007226880909081034, 3.86285069798194640589472842186, 5.10998135419359813983704925962, 6.13978679908661025711973561660, 7.002215624036116500276383598006, 7.98415858630547596323898186486, 8.4870235971484417578691056884, 9.278727606316618274433978778349, 10.2491723540680334390831448489, 11.06465332418579933458971842485, 11.75282019941227492266985728747, 12.64053798186832019333976641617, 13.16784208493272879024345624810, 15.06814932010144100674967287491, 15.22829620576538011481979021679, 15.98132138444886985409221255107, 16.67224270441540195841025851941, 17.75015407052966379101813036478, 18.45812877611766100786536830901, 18.90582924673209982720377965344, 19.87041993877837807459081145558, 20.42539045099848233706900361959, 21.089684881235091204678342051264

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