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Properties

Label 1-2001-2001.1172-r1-0-0
Degree $1$
Conductor $2001$
Sign $-0.189 + 0.981i$
Analytic cond. $215.037$
Root an. cond. $215.037$
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 − 4-s − 5-s − 7-s + i·8-s + i·10-s + i·11-s − 13-s + i·14-s + 16-s + i·17-s − i·19-s + 20-s + 22-s + 25-s + i·26-s + ⋯
L(s)  = 1  − i·2-s − 4-s − 5-s − 7-s + i·8-s + i·10-s + i·11-s − 13-s + i·14-s + 16-s + i·17-s − i·19-s + 20-s + 22-s + 25-s + i·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.189 + 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.189 + 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2001\)    =    \(3 \cdot 23 \cdot 29\)
Sign: $-0.189 + 0.981i$
Analytic conductor: \(215.037\)
Root analytic conductor: \(215.037\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2001} (1172, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2001,\ (1:\ ),\ -0.189 + 0.981i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3089218811 + 0.3740912908i\)
\(L(\frac12)\) \(\approx\) \(0.3089218811 + 0.3740912908i\)
\(L(1)\) \(\approx\) \(0.6048315582 - 0.1696044477i\)
\(L(1)\) \(\approx\) \(0.6048315582 - 0.1696044477i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
23 \( 1 \)
29 \( 1 \)
good2 \( 1 \)
5 \( 1 - iT \)
7 \( 1 \)
11 \( 1 - T \)
13 \( 1 - T \)
17 \( 1 \)
19 \( 1 - T \)
31 \( 1 + iT \)
37 \( 1 + iT \)
41 \( 1 \)
43 \( 1 - T \)
47 \( 1 + iT \)
53 \( 1 \)
59 \( 1 + T \)
61 \( 1 + iT \)
67 \( 1 \)
71 \( 1 - iT \)
73 \( 1 + T \)
79 \( 1 \)
83 \( 1 + T \)
89 \( 1 \)
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

−19.42255528200240940481611770285, −18.73931080022213204292735561769, −18.28025128417583272742186912346, −16.95734853836402818408872487169, −16.59667578291976979491561405324, −15.95005150154916157017905401154, −15.37633212387126537267972996885, −14.52004626286653997071261065808, −13.88744534230402233427012487629, −12.99899885378438582670504717722, −12.32507406304723605111502364249, −11.59138563058635672737545905166, −10.480515810471739184901465004492, −9.65633723959229995881870812369, −8.97650437564796784973959214126, −8.12502094293990128881079570318, −7.423440533201493534722158210038, −6.83529043956287725923956487558, −5.89169621247467847199765935399, −5.20404969111550858599524572178, −4.1172899854178477886817869876, −3.55547419356288004851854189355, −2.62099834202924235391449647628, −0.7089418482932845561022523903, −0.16792943486350505514676092193, 0.81373125349995201884246928661, 2.04440084573638544258124433717, 2.9314229091871682483003164299, 3.611530541829386294450991581768, 4.51930736554722254849275638985, 5.0421413209895453661377848104, 6.42334913612562656510953875358, 7.225207215311268141022065991347, 8.057668209870796125649748249830, 8.9573802680170005370352098681, 9.66529719762904798360804865937, 10.353198930459051606218077173, 11.077465769435799209344067755385, 12.074090634620683511319126422567, 12.42603750725071664054516466314, 13.022436997703294689667281346925, 13.96301314670341263383624670761, 15.03456076317667378956148207857, 15.30851966520097615971462660136, 16.44960396212588943775296309942, 17.2001658692680615583645948704, 17.88730086382248324992554200123, 18.90469546470058639900702444425, 19.34037103762486991310397296186, 20.01286475052549667335435645205

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