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Properties

Label 1-1001-1001.824-r1-0-0
Degree $1$
Conductor $1001$
Sign $-0.937 + 0.348i$
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  + (0.866 − 0.5i)2-s + (0.5 − 0.866i)3-s + (0.5 − 0.866i)4-s + (0.866 − 0.5i)5-s − i·6-s − i·8-s + (−0.5 − 0.866i)9-s + (0.5 − 0.866i)10-s + (−0.5 − 0.866i)12-s − i·15-s + (−0.5 − 0.866i)16-s + (0.5 − 0.866i)17-s + (−0.866 − 0.5i)18-s + (−0.866 + 0.5i)19-s − i·20-s + ⋯
L(s)  = 1  + (0.866 − 0.5i)2-s + (0.5 − 0.866i)3-s + (0.5 − 0.866i)4-s + (0.866 − 0.5i)5-s − i·6-s − i·8-s + (−0.5 − 0.866i)9-s + (0.5 − 0.866i)10-s + (−0.5 − 0.866i)12-s − i·15-s + (−0.5 − 0.866i)16-s + (0.5 − 0.866i)17-s + (−0.866 − 0.5i)18-s + (−0.866 + 0.5i)19-s − i·20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1001 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.937 + 0.348i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1001 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.937 + 0.348i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1001\)    =    \(7 \cdot 11 \cdot 13\)
Sign: $-0.937 + 0.348i$
Analytic conductor: \(107.572\)
Root analytic conductor: \(107.572\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1001} (824, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1001,\ (1:\ ),\ -0.937 + 0.348i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.8112721476 - 4.504974540i\)
\(L(\frac12)\) \(\approx\) \(-0.8112721476 - 4.504974540i\)
\(L(1)\) \(\approx\) \(1.375767756 - 1.747547325i\)
\(L(1)\) \(\approx\) \(1.375767756 - 1.747547325i\)

Euler product

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

−21.85927792834079589257931282194, −21.30383738803836973438319636668, −20.709532442455547667938875830180, −19.81156802312213016431071698684, −18.82399707411145208630169353716, −17.70341957447959293846213593442, −16.87391988021179568492992623032, −16.4073788894926412252524942500, −15.27039361530119187911449862921, −14.679830754688083397228445752993, −14.33577434351852890323408144917, −13.186495362346548657491733436779, −12.82547809335808860058854112688, −11.31811217226652754452722407835, −10.75278201102542107070747889376, −9.839961688988467480616021195498, −8.86901490133740471092749323247, −8.09175157344544180502843319456, −6.998910224877090803503767566541, −6.124263728811685910477499625187, −5.34272738150157243011907681428, −4.466088169426541029247983775423, −3.54334259270995729108927392905, −2.719937002262642553320117720399, −1.87573970368080503403849302747, 0.5582588969790846267331896447, 1.569872521714249434477220630316, 2.2435556230918317142928884187, 3.19294419025549352167001356879, 4.22617548954110739887498011742, 5.4585073674310877095272253972, 5.93781377037222084372365210802, 6.96538166983438822954977811684, 7.81077944642516292324181507310, 9.16436913455604577712491052874, 9.54374919811854515856522507297, 10.76069300676062317516474641727, 11.64508588895528059836453238577, 12.614997978161890965826642559698, 12.9632628563288427121661688335, 13.797394106064130972915637523988, 14.37714021193186107420386543109, 15.12249036569243259484516868585, 16.278511901868421055732876447255, 17.15618729623733547756642956142, 18.12659426580458841112694720780, 18.84473502520679281551819065058, 19.55326150256166586240837904182, 20.539387863469209010332312629306, 20.8153137020744262270324172447

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