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Properties

Label 1-4001-4001.1866-r0-0-0
Degree $1$
Conductor $4001$
Sign $0.964 - 0.265i$
Analytic cond. $18.5805$
Root an. cond. $18.5805$
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  − i·2-s + (−0.923 − 0.382i)3-s − 4-s − i·5-s + (−0.382 + 0.923i)6-s + i·7-s + i·8-s + (0.707 + 0.707i)9-s − 10-s + (−0.923 − 0.382i)11-s + (0.923 + 0.382i)12-s + i·13-s + 14-s + (−0.382 + 0.923i)15-s + 16-s + (0.923 − 0.382i)17-s + ⋯
L(s)  = 1  − i·2-s + (−0.923 − 0.382i)3-s − 4-s − i·5-s + (−0.382 + 0.923i)6-s + i·7-s + i·8-s + (0.707 + 0.707i)9-s − 10-s + (−0.923 − 0.382i)11-s + (0.923 + 0.382i)12-s + i·13-s + 14-s + (−0.382 + 0.923i)15-s + 16-s + (0.923 − 0.382i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.964 - 0.265i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.964 - 0.265i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4001\)
Sign: $0.964 - 0.265i$
Analytic conductor: \(18.5805\)
Root analytic conductor: \(18.5805\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4001} (1866, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4001,\ (0:\ ),\ 0.964 - 0.265i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6442873248 - 0.08722616493i\)
\(L(\frac12)\) \(\approx\) \(0.6442873248 - 0.08722616493i\)
\(L(1)\) \(\approx\) \(0.5585502573 - 0.3372420904i\)
\(L(1)\) \(\approx\) \(0.5585502573 - 0.3372420904i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad4001 \( 1 \)
good2 \( 1 - iT \)
3 \( 1 + (-0.923 - 0.382i)T \)
5 \( 1 - iT \)
7 \( 1 + iT \)
11 \( 1 + (-0.923 - 0.382i)T \)
13 \( 1 + iT \)
17 \( 1 + (0.923 - 0.382i)T \)
19 \( 1 + iT \)
23 \( 1 + (0.923 + 0.382i)T \)
29 \( 1 + (-0.707 - 0.707i)T \)
31 \( 1 + (0.707 - 0.707i)T \)
37 \( 1 + (-0.923 - 0.382i)T \)
41 \( 1 + (-0.382 - 0.923i)T \)
43 \( 1 + (-0.382 + 0.923i)T \)
47 \( 1 + (-0.707 - 0.707i)T \)
53 \( 1 + (0.923 - 0.382i)T \)
59 \( 1 + (0.707 + 0.707i)T \)
61 \( 1 + (-0.707 - 0.707i)T \)
67 \( 1 + (-0.923 + 0.382i)T \)
71 \( 1 + T \)
73 \( 1 + (-0.382 - 0.923i)T \)
79 \( 1 + iT \)
83 \( 1 + (-0.707 + 0.707i)T \)
89 \( 1 + (-0.707 - 0.707i)T \)
97 \( 1 + (0.707 + 0.707i)T \)
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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

−18.208232524622168279032467257119, −17.69625747072031784677671265631, −17.15324700716233029233366273722, −16.54722784746813145969599128985, −15.71462044481558268978766227338, −15.22243139758428606759081902883, −14.71873109973896434206171163373, −13.83377504709931038852419178508, −13.08080941008638227191263675733, −12.59677796115925978016659366855, −11.48619443407627755988186843891, −10.56509857416047563335743252461, −10.353511636752341030241127589634, −9.77245384946781610264441865662, −8.63084328956347591185447661456, −7.687269290017771194289695378855, −7.16410486158364393925629864712, −6.6850214811333854441779877928, −5.821135333234985955938613860784, −5.10230295904420307634192505037, −4.59473488010266030209389888274, −3.50192828855622402863297444227, −3.05123188127945301030061434125, −1.319904742671765500788668289434, −0.313690187144321690015275823229, 0.75027883049268336774914792064, 1.681024857603267701317690263086, 2.20092569669530742959460183075, 3.323088136650243994687501253008, 4.26023959980096032022282768573, 5.130717092867187683424138393635, 5.44451639580835395417567089098, 6.1101216796836004633260917517, 7.42533380615346224959613986395, 8.16272121853540150458904235409, 8.79356697957768962752409843065, 9.65443615737870694311836253529, 10.12684164664245581644742967386, 11.16430975124158200840086165103, 11.75675955348090492870164847211, 12.10028198643452366208426551763, 12.81514072985715756346727170735, 13.34125947522885908151645523924, 14.0156295746697312701223735215, 15.10135793554153040930563078520, 15.93589771963212674515701157516, 16.70016038792183105443733139503, 17.02399148099335180943491194730, 18.02593588826471276532916534330, 18.56450704059902338905066244779

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