↑

Properties

Label 1-1009-1009.540-r0-0-0
Degree $1$
Conductor $1009$
Sign $-0.857 - 0.513i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 3-s + 4-s − 5-s + 6-s + 7-s − 8-s + 9-s + 10-s − i·11-s − 12-s − i·13-s − 14-s + 15-s + 16-s − i·17-s + ⋯
L(s)  = 1  − 2-s − 3-s + 4-s − 5-s + 6-s + 7-s − 8-s + 9-s + 10-s − i·11-s − 12-s − i·13-s − 14-s + 15-s + 16-s − i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1009 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.857 - 0.513i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1009 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.857 - 0.513i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1009\)
Sign: $-0.857 - 0.513i$
Analytic conductor: \(4.68577\)
Root analytic conductor: \(4.68577\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1009} (540, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1009,\ (0:\ ),\ -0.857 - 0.513i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1060602641 - 0.3836014755i\)
\(L(\frac12)\) \(\approx\) \(0.1060602641 - 0.3836014755i\)
\(L(1)\) \(\approx\) \(0.4384922058 - 0.1459804637i\)
\(L(1)\) \(\approx\) \(0.4384922058 - 0.1459804637i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad1009 \( 1 \)
good2 \( 1 \)
3 \( 1 + T \)
5 \( 1 - T \)
7 \( 1 - T \)
11 \( 1 + T \)
13 \( 1 - T \)
17 \( 1 + T \)
19 \( 1 + T \)
23 \( 1 - T \)
29 \( 1 + T \)
31 \( 1 + T \)
37 \( 1 - iT \)
41 \( 1 - T \)
43 \( 1 - iT \)
47 \( 1 - T \)
53 \( 1 + T \)
59 \( 1 + T \)
61 \( 1 - iT \)
67 \( 1 - T \)
71 \( 1 - iT \)
73 \( 1 - T \)
79 \( 1 - T \)
83 \( 1 + iT \)
89 \( 1 - iT \)
97 \( 1 + T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−21.84126342870407261668637371388, −20.989917353536182453784067712724, −20.42716253564388532170278822644, −19.29876335797744336237166430509, −18.80626241253354461615645771729, −17.980829674195941073430467266918, −17.237935354324861370369484341296, −16.73864665195425724445717328498, −15.80426583985252081710481296477, −15.16501951347708950454607835198, −14.42973596929743106211950089755, −12.70507553092476816855191864930, −12.121898396925846850767641879929, −11.342018994621379595205709370622, −10.971795100626619829023649795615, −9.95392079964612992895134455938, −9.103648021471892549184931108735, −7.82728889333384053574776193469, −7.5987377088648446952651298690, −6.586630929119525024509595723595, −5.63291383246694500825030176636, −4.494896156001170177757573851165, −3.76739040753177269285427001898, −1.93543750227472833556924257059, −1.35887162183405086612760412673, 0.34116580003575680164431945059, 1.061689592899598207067697765296, 2.54001173744292029349461966813, 3.685405897800278563753697000142, 4.95215791541714062356980193296, 5.61603941709582862487656133380, 6.88713211598221724694172107179, 7.41148469273799969943223190235, 8.34398010523105437396014901932, 8.980570862118691704961507966, 10.425422993635854720244319398039, 10.84328505017847674987981948074, 11.56729211057008761705059549365, 12.04792286222637329083530274861, 13.1097709122637060928369438367, 14.49110135108557079888021667268, 15.40693130940450919846703678875, 15.96780552445298999423747839530, 16.66512677835495272254919274152, 17.48515708330760791305304183849, 18.18559751121301775644367988715, 18.71404517786265650534403623838, 19.61821721706039064839029831683, 20.46695813987719498336258590254, 21.127581034652222121679718713067

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