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Properties

Label 1-1040-1040.883-r0-0-0
Degree $1$
Conductor $1040$
Sign $0.987 - 0.160i$
Analytic cond. $4.82973$
Root an. cond. $4.82973$
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  + 3-s + i·7-s + 9-s − i·11-s − i·17-s − i·19-s + i·21-s + i·23-s + 27-s − i·29-s + 31-s − i·33-s + 37-s + 41-s − 43-s + ⋯
L(s)  = 1  + 3-s + i·7-s + 9-s − i·11-s − i·17-s − i·19-s + i·21-s + i·23-s + 27-s − i·29-s + 31-s − i·33-s + 37-s + 41-s − 43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.987 - 0.160i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.987 - 0.160i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1040\)    =    \(2^{4} \cdot 5 \cdot 13\)
Sign: $0.987 - 0.160i$
Analytic conductor: \(4.82973\)
Root analytic conductor: \(4.82973\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1040} (883, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1040,\ (0:\ ),\ 0.987 - 0.160i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.281672282 - 0.1839291887i\)
\(L(\frac12)\) \(\approx\) \(2.281672282 - 0.1839291887i\)
\(L(1)\) \(\approx\) \(1.551233949 + 0.02323274606i\)
\(L(1)\) \(\approx\) \(1.551233949 + 0.02323274606i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
13 \( 1 \)
good3 \( 1 + T \)
7 \( 1 + T \)
11 \( 1 \)
17 \( 1 \)
19 \( 1 + iT \)
23 \( 1 \)
29 \( 1 + T \)
31 \( 1 \)
37 \( 1 - iT \)
41 \( 1 \)
43 \( 1 \)
47 \( 1 \)
53 \( 1 \)
59 \( 1 \)
61 \( 1 - iT \)
67 \( 1 \)
71 \( 1 - iT \)
73 \( 1 \)
79 \( 1 + iT \)
83 \( 1 \)
89 \( 1 + iT \)
97 \( 1 \)
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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

−21.331505089234434368610413990488, −20.63275614381886515871793545651, −20.08760393980010736870974222413, −19.43137907075701413723465255589, −18.56033489630274752320122690653, −17.73449845702469007977109977568, −16.83004394729592910322585370768, −16.08866913814493004658407152462, −15.03040888297348472289093013520, −14.525458035813162170925306633145, −13.81874437495644170037732716994, −12.80973535263097283538529859426, −12.43759484330435287294215812672, −10.978485894876662243919736888628, −10.16556821863861046862091206031, −9.681542287318824331632074946, −8.497257112341827524467971064799, −7.87654247989907060904409252946, −7.0666154992121248970306415911, −6.247476337236136224291289916877, −4.70098821038582683204349193625, −4.12031931373105191501544802335, −3.22610241687314411462391513054, −2.091253761906776904219266984492, −1.215028340187065418823749294720, 1.004219966186519188076911955761, 2.43595255263678388689750462732, 2.84699462661456662215378127403, 3.94227673944745454165499095269, 5.01779901143843404223424612148, 5.94861301725113122227658944348, 6.98671172988600090832629911624, 7.92410947581891714912340540027, 8.691269503634920604885403807509, 9.30060572644132966464166754434, 10.070773447179682735132438712029, 11.38215285095658896888427455320, 11.87062589597777727240265852923, 13.169430455630153600780323734137, 13.544545789149596282745351962118, 14.445096116752055405086950894845, 15.36964067109597259018858646770, 15.74180906285706973090123945135, 16.70040780883742635233522199866, 17.94039921828441717489337456271, 18.48614080103819540800748217810, 19.34783871566238776979112420759, 19.7716932680937442060771324983, 20.90228081856264047798805558023, 21.445035957955263063580625310737

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