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Properties

Label 1-1009-1009.105-r0-0-0
Degree $1$
Conductor $1009$
Sign $-0.979 + 0.199i$
Analytic cond. $4.68577$
Root an. cond. $4.68577$
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  + (−0.222 + 0.974i)2-s + (−0.222 + 0.974i)3-s + (−0.900 − 0.433i)4-s + (−0.900 − 0.433i)5-s + (−0.900 − 0.433i)6-s + (−0.222 + 0.974i)7-s + (0.623 − 0.781i)8-s + (−0.900 − 0.433i)9-s + (0.623 − 0.781i)10-s + (−0.900 − 0.433i)11-s + (0.623 − 0.781i)12-s + 13-s + (−0.900 − 0.433i)14-s + (0.623 − 0.781i)15-s + (0.623 + 0.781i)16-s + (0.623 + 0.781i)17-s + ⋯
L(s)  = 1  + (−0.222 + 0.974i)2-s + (−0.222 + 0.974i)3-s + (−0.900 − 0.433i)4-s + (−0.900 − 0.433i)5-s + (−0.900 − 0.433i)6-s + (−0.222 + 0.974i)7-s + (0.623 − 0.781i)8-s + (−0.900 − 0.433i)9-s + (0.623 − 0.781i)10-s + (−0.900 − 0.433i)11-s + (0.623 − 0.781i)12-s + 13-s + (−0.900 − 0.433i)14-s + (0.623 − 0.781i)15-s + (0.623 + 0.781i)16-s + (0.623 + 0.781i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1009 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.979 + 0.199i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1009 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.979 + 0.199i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1009\)
Sign: $-0.979 + 0.199i$
Analytic conductor: \(4.68577\)
Root analytic conductor: \(4.68577\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1009} (105, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1009,\ (0:\ ),\ -0.979 + 0.199i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.07638534279 + 0.7562739605i\)
\(L(\frac12)\) \(\approx\) \(0.07638534279 + 0.7562739605i\)
\(L(1)\) \(\approx\) \(0.4722300648 + 0.5008112433i\)
\(L(1)\) \(\approx\) \(0.4722300648 + 0.5008112433i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad1009 \( 1 \)
good2 \( 1 + (-0.222 + 0.974i)T \)
3 \( 1 + (-0.222 + 0.974i)T \)
5 \( 1 + (-0.900 - 0.433i)T \)
7 \( 1 + (-0.222 + 0.974i)T \)
11 \( 1 + (-0.900 - 0.433i)T \)
13 \( 1 + T \)
17 \( 1 + (0.623 + 0.781i)T \)
19 \( 1 + T \)
23 \( 1 + (0.623 + 0.781i)T \)
29 \( 1 + (-0.222 - 0.974i)T \)
31 \( 1 + (-0.222 - 0.974i)T \)
37 \( 1 + T \)
41 \( 1 + T \)
43 \( 1 + T \)
47 \( 1 + (-0.222 + 0.974i)T \)
53 \( 1 + (-0.222 + 0.974i)T \)
59 \( 1 + (-0.900 + 0.433i)T \)
61 \( 1 + (-0.222 - 0.974i)T \)
67 \( 1 + (-0.222 - 0.974i)T \)
71 \( 1 + (-0.222 + 0.974i)T \)
73 \( 1 + (-0.900 + 0.433i)T \)
79 \( 1 + (0.623 - 0.781i)T \)
83 \( 1 + (0.623 + 0.781i)T \)
89 \( 1 + (-0.900 - 0.433i)T \)
97 \( 1 + (-0.900 + 0.433i)T \)
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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

−20.87963751190541929962863380544, −20.36782953888838884059953017317, −19.68487843922607125958079093278, −18.943519061876464605200078856616, −18.14092688751171833930328919721, −17.963911129369189771099642154590, −16.54963890229362461464141155981, −16.13185780070945606908122079600, −14.58391582792507758690975879752, −13.90657377034809435138940882373, −13.09130476345841989376546585106, −12.49048812764074798228883365066, −11.61800457949669778584930288734, −10.897233594952067485903719072762, −10.40898947202520217891468232510, −9.15285103042237448286855939537, −8.09516997176217560725653880807, −7.50650914187686899265283830880, −6.87148920060605575837808364328, −5.430617709263059249782162911798, −4.45325098176916967747592582746, −3.30074834044189638987049137480, −2.79727224744464912912766157397, −1.349316260060182225249756056539, −0.528859327574244609681694696588, 0.94352079666035231879546409885, 3.01267435071637028742827446889, 3.85690375151628416017976903140, 4.74679331022605905393953790183, 5.79689139563445706209855769153, 5.91959431009188449675139072052, 7.68367658674653296323627664793, 8.121143617397358786462719779109, 9.11160702266792800220751508445, 9.542996511450378047670198737, 10.786637659964926151309058466, 11.43271034037420581951006007994, 12.539647083529975912042345575524, 13.363095554852046425946953541066, 14.46833127023776722838648476906, 15.47296123612203800847193356019, 15.5606749646713601982676969293, 16.29732840899060982661304481126, 16.98267610682054908215335829252, 18.02429017997964911947392649083, 18.795531297590019729094982230637, 19.43354928561430734138243469060, 20.61072426991289390134187220671, 21.28264332289190131804459250989, 22.13090150318824473171346407445

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