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Properties

Label 1-1001-1001.219-r0-0-0
Degree $1$
Conductor $1001$
Sign $-0.994 - 0.101i$
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  − i·2-s + (−0.5 − 0.866i)3-s − 4-s + (0.866 − 0.5i)5-s + (−0.866 + 0.5i)6-s + i·8-s + (−0.5 + 0.866i)9-s + (−0.5 − 0.866i)10-s + (0.5 + 0.866i)12-s + (−0.866 − 0.5i)15-s + 16-s + 17-s + (0.866 + 0.5i)18-s + (0.866 + 0.5i)19-s + (−0.866 + 0.5i)20-s + ⋯
L(s)  = 1  − i·2-s + (−0.5 − 0.866i)3-s − 4-s + (0.866 − 0.5i)5-s + (−0.866 + 0.5i)6-s + i·8-s + (−0.5 + 0.866i)9-s + (−0.5 − 0.866i)10-s + (0.5 + 0.866i)12-s + (−0.866 − 0.5i)15-s + 16-s + 17-s + (0.866 + 0.5i)18-s + (0.866 + 0.5i)19-s + (−0.866 + 0.5i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.994 - 0.101i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.994 - 0.101i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1001\)    =    \(7 \cdot 11 \cdot 13\)
Sign: $-0.994 - 0.101i$
Analytic conductor: \(4.64862\)
Root analytic conductor: \(4.64862\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1001} (219, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1001,\ (0:\ ),\ -0.994 - 0.101i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.06145664238 - 1.204117896i\)
\(L(\frac12)\) \(\approx\) \(0.06145664238 - 1.204117896i\)
\(L(1)\) \(\approx\) \(0.6003641535 - 0.7427324140i\)
\(L(1)\) \(\approx\) \(0.6003641535 - 0.7427324140i\)

Euler product

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

−22.0686508840115706396952167814, −21.61238448453174087873684397729, −20.75995213866729588807537251217, −19.65405767907665050876079588270, −18.398383669745772859827279234959, −17.960288797331862913841823417565, −17.26865533395530116207543701950, −16.365153001336214596873508143504, −15.98415750156332114172565921468, −14.83533253568769697948411882408, −14.428831150539724411852775535893, −13.62732866129209557973441874274, −12.59934187585172168611728828404, −11.58789891537003608407783041082, −10.46543351208130523172239170421, −9.868459583132246945353210651904, −9.25566599673567667973780188409, −8.24858315221324771312788767513, −7.10009878632341801908751566213, −6.358462115981655178777450863669, −5.48741376846479273920865885798, −5.03665261110787580334940264831, −3.80652118903811799115613571942, −2.97411106479968885233935823081, −1.19563173083762371717298569714, 0.63868423380858458484238165704, 1.61421780956641702569183642186, 2.288818106770639050577477247509, 3.46871758081921488514903620916, 4.73627652977389997515853207228, 5.58634785144078143016653040265, 6.13633401946132554198348161442, 7.6160650624097910508905615640, 8.26018992023067471313018638897, 9.42643592244971809899398326734, 10.00309487070586264950032665663, 10.955985098494690651727439159377, 11.85271189455539512931243788512, 12.46964997791345950522574683117, 13.080696791854013256073489723474, 14.01599135844252291906752643228, 14.31464008524531822963080247752, 16.08718314060703959330478688479, 16.896865668790945125776771043175, 17.56944951565173270360461390614, 18.2659579201959638493858949044, 18.809918140522822839080149585497, 19.79154476589974114360273948323, 20.43355438090803913549049511637, 21.28140469695338948151972618159

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