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Properties

Label 1-1009-1009.103-r0-0-0
Degree $1$
Conductor $1009$
Sign $0.0498 + 0.998i$
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  + (−0.997 − 0.0747i)2-s + (0.433 + 0.900i)3-s + (0.988 + 0.149i)4-s + (0.930 − 0.365i)5-s + (−0.365 − 0.930i)6-s + (−0.826 − 0.563i)7-s + (−0.974 − 0.222i)8-s + (−0.623 + 0.781i)9-s + (−0.955 + 0.294i)10-s + (−0.399 + 0.916i)11-s + (0.294 + 0.955i)12-s + (−0.707 − 0.707i)13-s + (0.781 + 0.623i)14-s + (0.733 + 0.680i)15-s + (0.955 + 0.294i)16-s + (0.467 + 0.884i)17-s + ⋯
L(s)  = 1  + (−0.997 − 0.0747i)2-s + (0.433 + 0.900i)3-s + (0.988 + 0.149i)4-s + (0.930 − 0.365i)5-s + (−0.365 − 0.930i)6-s + (−0.826 − 0.563i)7-s + (−0.974 − 0.222i)8-s + (−0.623 + 0.781i)9-s + (−0.955 + 0.294i)10-s + (−0.399 + 0.916i)11-s + (0.294 + 0.955i)12-s + (−0.707 − 0.707i)13-s + (0.781 + 0.623i)14-s + (0.733 + 0.680i)15-s + (0.955 + 0.294i)16-s + (0.467 + 0.884i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1009 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0498 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1009 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0498 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1009\)
Sign: $0.0498 + 0.998i$
Analytic conductor: \(4.68577\)
Root analytic conductor: \(4.68577\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1009} (103, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1009,\ (0:\ ),\ 0.0498 + 0.998i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7457597989 + 0.7094341466i\)
\(L(\frac12)\) \(\approx\) \(0.7457597989 + 0.7094341466i\)
\(L(1)\) \(\approx\) \(0.7895166189 + 0.2548279817i\)
\(L(1)\) \(\approx\) \(0.7895166189 + 0.2548279817i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad1009 \( 1 \)
good2 \( 1 + (-0.997 - 0.0747i)T \)
3 \( 1 + (0.433 + 0.900i)T \)
5 \( 1 + (0.930 - 0.365i)T \)
7 \( 1 + (-0.826 - 0.563i)T \)
11 \( 1 + (-0.399 + 0.916i)T \)
13 \( 1 + (-0.707 - 0.707i)T \)
17 \( 1 + (0.467 + 0.884i)T \)
19 \( 1 + (0.707 + 0.707i)T \)
23 \( 1 + (0.532 - 0.846i)T \)
29 \( 1 + (0.563 + 0.826i)T \)
31 \( 1 + (-0.757 - 0.652i)T \)
37 \( 1 + (0.866 - 0.5i)T \)
41 \( 1 + (0.5 + 0.866i)T \)
43 \( 1 + (0.707 + 0.707i)T \)
47 \( 1 + (0.943 + 0.330i)T \)
53 \( 1 + (-0.757 + 0.652i)T \)
59 \( 1 + (-0.993 - 0.111i)T \)
61 \( 1 + (-0.943 + 0.330i)T \)
67 \( 1 + (0.826 - 0.563i)T \)
71 \( 1 + (0.997 + 0.0747i)T \)
73 \( 1 + (-0.993 - 0.111i)T \)
79 \( 1 + (0.0373 + 0.999i)T \)
83 \( 1 + (-0.884 + 0.467i)T \)
89 \( 1 + (-0.399 + 0.916i)T \)
97 \( 1 + (-0.593 + 0.804i)T \)
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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

−21.38014518277371621653455709482, −20.452006094204578138466163725275, −19.55202793749629801320382640807, −18.88767341389030476861799703223, −18.5358045436569706841144931977, −17.68141930250376476456030684910, −16.98511692944429916958850452201, −16.06757992905477801191491911284, −15.2604530793160695825251800746, −14.16757466006878362712438834290, −13.657968231193999939229032410986, −12.63445448312213635662223388738, −11.801803317765843401433500838354, −11.01461208666668070731378242836, −9.75793977379255642463661211570, −9.34121510152061496213113995803, −8.65837864700722021542185719018, −7.41369912482307981363563366296, −6.98026153054411049334143613830, −6.02038094350043719621420133332, −5.4385497519771463418743599920, −3.087354684747366760497839921954, −2.78205065260914704956383598762, −1.8195435559349288150654851649, −0.60973317018498146127980011105, 1.1611282140226911923614770996, 2.42751882523448616870162236622, 3.06349016958368906396852671687, 4.28927444270059013848572030277, 5.424998139957357626079961337141, 6.244331343126460368279390788209, 7.45633593792602476345229867117, 8.094520171557178824090371423325, 9.36604172881400254607727131090, 9.57504735620019404566451920782, 10.40377187114413903268860043987, 10.78961751802861138143558678465, 12.52473347384236386264293050931, 12.7648387053055540060919360254, 14.16177659047291908903847162183, 14.851270600090760253340400869351, 15.72554096649916782709057166277, 16.61587657122456294108400211657, 16.918035682193298099516460037240, 17.81099260327898169890738595473, 18.6742568690598208543938060429, 19.79982083570047317876313490468, 20.156183371046959825765751835172, 20.806812196581977541755689006910, 21.59301579601169715708152330150

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