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Properties

Label 1-1001-1001.752-r0-0-0
Degree $1$
Conductor $1001$
Sign $-0.388 + 0.921i$
Analytic cond. $4.64862$
Root an. cond. $4.64862$
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  + (0.743 + 0.669i)2-s + (0.809 + 0.587i)3-s + (0.104 + 0.994i)4-s + (0.743 − 0.669i)5-s + (0.207 + 0.978i)6-s + (−0.587 + 0.809i)8-s + (0.309 + 0.951i)9-s + 10-s + (−0.5 + 0.866i)12-s + (0.994 − 0.104i)15-s + (−0.978 + 0.207i)16-s + (0.669 + 0.743i)17-s + (−0.406 + 0.913i)18-s + (−0.587 + 0.809i)19-s + (0.743 + 0.669i)20-s + ⋯
L(s)  = 1  + (0.743 + 0.669i)2-s + (0.809 + 0.587i)3-s + (0.104 + 0.994i)4-s + (0.743 − 0.669i)5-s + (0.207 + 0.978i)6-s + (−0.587 + 0.809i)8-s + (0.309 + 0.951i)9-s + 10-s + (−0.5 + 0.866i)12-s + (0.994 − 0.104i)15-s + (−0.978 + 0.207i)16-s + (0.669 + 0.743i)17-s + (−0.406 + 0.913i)18-s + (−0.587 + 0.809i)19-s + (0.743 + 0.669i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.388 + 0.921i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.388 + 0.921i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1001\)    =    \(7 \cdot 11 \cdot 13\)
Sign: $-0.388 + 0.921i$
Analytic conductor: \(4.64862\)
Root analytic conductor: \(4.64862\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1001} (752, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1001,\ (0:\ ),\ -0.388 + 0.921i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.872488849 + 2.820287537i\)
\(L(\frac12)\) \(\approx\) \(1.872488849 + 2.820287537i\)
\(L(1)\) \(\approx\) \(1.778489658 + 1.306700805i\)
\(L(1)\) \(\approx\) \(1.778489658 + 1.306700805i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
11 \( 1 \)
13 \( 1 \)
good2 \( 1 + (0.743 + 0.669i)T \)
3 \( 1 + (0.809 + 0.587i)T \)
5 \( 1 + (0.743 - 0.669i)T \)
17 \( 1 + (0.669 + 0.743i)T \)
19 \( 1 + (-0.587 + 0.809i)T \)
23 \( 1 + (0.5 + 0.866i)T \)
29 \( 1 + (-0.104 - 0.994i)T \)
31 \( 1 + (-0.743 - 0.669i)T \)
37 \( 1 + (0.406 - 0.913i)T \)
41 \( 1 + (-0.406 - 0.913i)T \)
43 \( 1 + (0.5 + 0.866i)T \)
47 \( 1 + (0.406 + 0.913i)T \)
53 \( 1 + (0.669 - 0.743i)T \)
59 \( 1 + (0.994 - 0.104i)T \)
61 \( 1 + (-0.309 + 0.951i)T \)
67 \( 1 + iT \)
71 \( 1 + (-0.207 - 0.978i)T \)
73 \( 1 + (-0.406 + 0.913i)T \)
79 \( 1 + (-0.978 - 0.207i)T \)
83 \( 1 + (-0.951 - 0.309i)T \)
89 \( 1 + (0.866 - 0.5i)T \)
97 \( 1 + (-0.743 - 0.669i)T \)
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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

−21.48133728141076557125257775011, −20.55170568053887355890261012989, −20.05670983728141668016548707874, −19.051152017877021587192492458690, −18.5252092916874284668919311480, −17.93533014949605917713766112450, −16.71294304375235857826606836777, −15.42802760629782556308897587005, −14.73677088871735975249943180527, −14.193573742194181610069405642437, −13.46796381762353105432633574967, −12.842191488008833980229402876546, −12.01241135676977009832246976006, −11.00329557151441510120739169024, −10.2255430618550845116677480673, −9.387619298686950454797711149913, −8.64225219857779511677021115401, −7.14873402320862617219253984655, −6.72843748046374162576797259017, −5.69481063920535414659233076854, −4.70946424038714748987090455583, −3.45033088918210671658005044975, −2.81415440365290138321228550143, −2.06990073390089260185032502196, −1.044391875600966845648892407, 1.70806652227336867637858685272, 2.62400074956362727739198595917, 3.78863224697101228895588277644, 4.33847570143869037002088566139, 5.50089774754676946219966657067, 5.91403257034717179874064487373, 7.31671130360301302779601525175, 8.08515795423200311648815542632, 8.839250714773961770387362079689, 9.610587082316448964829123658634, 10.520413406182038125049608012164, 11.706380580936786501362949684276, 12.889539853069452100427884203589, 13.13626977335802421330848542860, 14.2027760168455042565923910826, 14.64102501705491279842009507917, 15.51162466882471301915605351983, 16.33079739423422932911090859808, 16.92713661069514477604281490989, 17.63580074914552456269330036488, 18.87659546423681013483853232974, 19.77019120040485504784144107507, 20.8938381470363629513020955506, 20.99475685546493055762668913927, 21.772152820739030553749490556689

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