Properties

Label 1-1001-1001.142-r1-0-0
Degree $1$
Conductor $1001$
Sign $-0.0633 + 0.997i$
Analytic cond. $107.572$
Root an. cond. $107.572$
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  + (−0.5 − 0.866i)2-s + (−0.5 + 0.866i)3-s + (−0.5 + 0.866i)4-s + (0.5 + 0.866i)5-s + 6-s + 8-s + (−0.5 − 0.866i)9-s + (0.5 − 0.866i)10-s + (−0.5 − 0.866i)12-s − 15-s + (−0.5 − 0.866i)16-s + (0.5 − 0.866i)17-s + (−0.5 + 0.866i)18-s + (−0.5 − 0.866i)19-s − 20-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)2-s + (−0.5 + 0.866i)3-s + (−0.5 + 0.866i)4-s + (0.5 + 0.866i)5-s + 6-s + 8-s + (−0.5 − 0.866i)9-s + (0.5 − 0.866i)10-s + (−0.5 − 0.866i)12-s − 15-s + (−0.5 − 0.866i)16-s + (0.5 − 0.866i)17-s + (−0.5 + 0.866i)18-s + (−0.5 − 0.866i)19-s − 20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1001 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.0633 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1001 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.0633 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1001\)    =    \(7 \cdot 11 \cdot 13\)
Sign: $-0.0633 + 0.997i$
Analytic conductor: \(107.572\)
Root analytic conductor: \(107.572\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1001} (142, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1001,\ (1:\ ),\ -0.0633 + 0.997i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5742575576 + 0.6118552101i\)
\(L(\frac12)\) \(\approx\) \(0.5742575576 + 0.6118552101i\)
\(L(1)\) \(\approx\) \(0.6865635052 + 0.04356944686i\)
\(L(1)\) \(\approx\) \(0.6865635052 + 0.04356944686i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
11 \( 1 \)
13 \( 1 \)
good2 \( 1 + (-0.5 - 0.866i)T \)
3 \( 1 + (-0.5 + 0.866i)T \)
5 \( 1 + (0.5 + 0.866i)T \)
17 \( 1 + (0.5 - 0.866i)T \)
19 \( 1 + (-0.5 - 0.866i)T \)
23 \( 1 + (-0.5 - 0.866i)T \)
29 \( 1 - T \)
31 \( 1 + (0.5 - 0.866i)T \)
37 \( 1 + (0.5 + 0.866i)T \)
41 \( 1 + T \)
43 \( 1 - T \)
47 \( 1 + (0.5 + 0.866i)T \)
53 \( 1 + (-0.5 + 0.866i)T \)
59 \( 1 + (0.5 - 0.866i)T \)
61 \( 1 + (0.5 + 0.866i)T \)
67 \( 1 + (0.5 - 0.866i)T \)
71 \( 1 - T \)
73 \( 1 + (-0.5 + 0.866i)T \)
79 \( 1 + (0.5 + 0.866i)T \)
83 \( 1 + T \)
89 \( 1 + (0.5 + 0.866i)T \)
97 \( 1 - T \)
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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

−21.35351774268961610722796279729, −20.22545938846771106109843465410, −19.40839323706574153107759522839, −18.81676456099975143707715875173, −17.77619221718061550977486643808, −17.4758647883704808755093709557, −16.54442799482449174347182874273, −16.2146104267855718385365659828, −14.91581985280923540964580563787, −14.13397350947176345366071411463, −13.28828316408141636256624271343, −12.70140169663563608925332554293, −11.770140517843409539934191901260, −10.61453518128800104830418729981, −9.877280248302634338409162474296, −8.84750496246079405119669375693, −8.12671042892860585926304918809, −7.456238858208171979732340409300, −6.33998051236289642492332487561, −5.76831086930719222548682094531, −5.11458837675224417330118616488, −3.93701177927002138072774657889, −1.982322373578656867271046516201, −1.38614142629996933389002392008, −0.29758026654051886238906163793, 0.82258401926083778876526792036, 2.33030471653957004327785023216, 3.005671668636694112337647496823, 4.01956869589034731465461559650, 4.8742530982244500593571781631, 5.97837942573867892971773598022, 6.92827974524686187761917324448, 8.00223334804977452764026748676, 9.17039715035595981563080080018, 9.706256177990604546973818393927, 10.42422564117617718757035220883, 11.15886217962671754757533801791, 11.67880505420953735515363004496, 12.734564785761577592026169370575, 13.70113048910492902521389597713, 14.53085932304306552556091861524, 15.40249412777612932896084927822, 16.42674762542259540004448282785, 17.08605382911145533198032219506, 17.83292947379950747604590827385, 18.48901424123726938309418624297, 19.223169468586221048822765045211, 20.40179229239746306976803769643, 20.79476578210322040542139608630, 21.73704803787126842336362071813

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