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Properties

Label 1-1009-1009.112-r0-0-0
Degree $1$
Conductor $1009$
Sign $-0.904 + 0.426i$
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  + (−0.365 − 0.930i)2-s + (−0.623 + 0.781i)3-s + (−0.733 + 0.680i)4-s + (−0.955 − 0.294i)5-s + (0.955 + 0.294i)6-s + (−0.988 − 0.149i)7-s + (0.900 + 0.433i)8-s + (−0.222 − 0.974i)9-s + (0.0747 + 0.997i)10-s + (−0.294 + 0.955i)11-s + (−0.0747 − 0.997i)12-s + i·13-s + (0.222 + 0.974i)14-s + (0.826 − 0.563i)15-s + (0.0747 − 0.997i)16-s + (0.997 + 0.0747i)17-s + ⋯
L(s)  = 1  + (−0.365 − 0.930i)2-s + (−0.623 + 0.781i)3-s + (−0.733 + 0.680i)4-s + (−0.955 − 0.294i)5-s + (0.955 + 0.294i)6-s + (−0.988 − 0.149i)7-s + (0.900 + 0.433i)8-s + (−0.222 − 0.974i)9-s + (0.0747 + 0.997i)10-s + (−0.294 + 0.955i)11-s + (−0.0747 − 0.997i)12-s + i·13-s + (0.222 + 0.974i)14-s + (0.826 − 0.563i)15-s + (0.0747 − 0.997i)16-s + (0.997 + 0.0747i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1009 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.904 + 0.426i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1009 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.904 + 0.426i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1009\)
Sign: $-0.904 + 0.426i$
Analytic conductor: \(4.68577\)
Root analytic conductor: \(4.68577\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1009} (112, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1009,\ (0:\ ),\ -0.904 + 0.426i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.03681438747 + 0.1645448321i\)
\(L(\frac12)\) \(\approx\) \(0.03681438747 + 0.1645448321i\)
\(L(1)\) \(\approx\) \(0.4501116930 + 0.006434569639i\)
\(L(1)\) \(\approx\) \(0.4501116930 + 0.006434569639i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad1009 \( 1 \)
good2 \( 1 + (-0.365 - 0.930i)T \)
3 \( 1 + (-0.623 + 0.781i)T \)
5 \( 1 + (-0.955 - 0.294i)T \)
7 \( 1 + (-0.988 - 0.149i)T \)
11 \( 1 + (-0.294 + 0.955i)T \)
13 \( 1 + iT \)
17 \( 1 + (0.997 + 0.0747i)T \)
19 \( 1 + iT \)
23 \( 1 + (-0.433 - 0.900i)T \)
29 \( 1 + (0.988 - 0.149i)T \)
31 \( 1 + (-0.930 - 0.365i)T \)
37 \( 1 + (0.5 + 0.866i)T \)
41 \( 1 + (-0.5 - 0.866i)T \)
43 \( 1 + iT \)
47 \( 1 + (0.781 + 0.623i)T \)
53 \( 1 + (-0.930 + 0.365i)T \)
59 \( 1 + (0.974 + 0.222i)T \)
61 \( 1 + (0.781 - 0.623i)T \)
67 \( 1 + (-0.988 + 0.149i)T \)
71 \( 1 + (-0.365 - 0.930i)T \)
73 \( 1 + (0.974 + 0.222i)T \)
79 \( 1 + (0.563 + 0.826i)T \)
83 \( 1 + (-0.997 - 0.0747i)T \)
89 \( 1 + (-0.294 + 0.955i)T \)
97 \( 1 + (-0.680 + 0.733i)T \)
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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

−21.813286164746582377445358983906, −19.93510330216259898703303705803, −19.49676460668130911465321752683, −18.810408279399726410764235714129, −18.20514337914047574711926407087, −17.40646658896243732287082915496, −16.367927341737116616041135126, −16.04147595204735707442760386379, −15.27030640108377910669364491801, −14.20454687346864022668048884929, −13.33360424319788114797869495634, −12.699212029001619398936272745003, −11.73529406176342189421291105973, −10.786320879253400986023671780705, −10.10084453174486366175745896992, −8.8397081281616459167655704063, −8.04000847955662554309891873419, −7.36835464205464022438811196243, −6.67062799404390976515854106514, −5.75066731370336918583413535927, −5.196011720628726474384223170465, −3.73314146601473691501506725482, −2.80274066840938077834941761264, −0.96138221773224153261535609500, −0.128689583722390163059431100388, 1.21207186698073849472305338348, 2.72513528338124532500143158386, 3.77178835237648929425167556398, 4.20718290908145293687895694604, 5.110734607707622071975655253680, 6.40677037399424805132173351895, 7.45382248327110724420405633780, 8.41269885560563277899653292298, 9.42558347835095307048301330853, 9.954096912999998167094937670757, 10.67105644902607615916357564098, 11.63488470322576554862665893746, 12.37177195394119925298478024957, 12.59917646391853736313784343952, 14.047063423899817174755819297612, 14.95895139873701161793846168159, 16.06474819327371267862435456391, 16.491520653377567459417790110509, 17.105705259599311592277900345121, 18.286866567451092214346038398389, 18.91050326983934830892251175365, 19.69706573214549849607528019831, 20.58827293919452899839727814664, 20.88097792675655139087090163818, 22.05831036242405303542905433428

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