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Properties

Label 1-1001-1001.571-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: $\chi_{1001} (571, \cdot )$
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

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

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