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Properties

Label 1-1001-1001.174-r0-0-0
Degree $1$
Conductor $1001$
Sign $0.958 - 0.284i$
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  + (−0.587 + 0.809i)2-s + (−0.309 + 0.951i)3-s + (−0.309 − 0.951i)4-s + (−0.587 − 0.809i)5-s + (−0.587 − 0.809i)6-s + (0.951 + 0.309i)8-s + (−0.809 − 0.587i)9-s + 10-s + 12-s + (0.951 − 0.309i)15-s + (−0.809 + 0.587i)16-s + (−0.809 + 0.587i)17-s + (0.951 − 0.309i)18-s + (0.951 + 0.309i)19-s + (−0.587 + 0.809i)20-s + ⋯
L(s)  = 1  + (−0.587 + 0.809i)2-s + (−0.309 + 0.951i)3-s + (−0.309 − 0.951i)4-s + (−0.587 − 0.809i)5-s + (−0.587 − 0.809i)6-s + (0.951 + 0.309i)8-s + (−0.809 − 0.587i)9-s + 10-s + 12-s + (0.951 − 0.309i)15-s + (−0.809 + 0.587i)16-s + (−0.809 + 0.587i)17-s + (0.951 − 0.309i)18-s + (0.951 + 0.309i)19-s + (−0.587 + 0.809i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.958 - 0.284i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.958 - 0.284i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1001\)    =    \(7 \cdot 11 \cdot 13\)
Sign: $0.958 - 0.284i$
Analytic conductor: \(4.64862\)
Root analytic conductor: \(4.64862\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1001} (174, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1001,\ (0:\ ),\ 0.958 - 0.284i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4529322907 - 0.06580577657i\)
\(L(\frac12)\) \(\approx\) \(0.4529322907 - 0.06580577657i\)
\(L(1)\) \(\approx\) \(0.5111655624 + 0.2090490624i\)
\(L(1)\) \(\approx\) \(0.5111655624 + 0.2090490624i\)

Euler product

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

−21.80080144974014214509166734977, −20.67504249006484469812962763573, −19.75794447741600400067141248427, −19.42136479920251015637190856895, −18.54915349376480567474416824265, −17.91339524731344692161974040863, −17.51419686509238809562370241485, −16.266662379749688816547171483722, −15.64173434015543901451098539849, −14.17273382966505421373139291576, −13.7029819240345411986160712415, −12.70536174659777916393332553068, −11.801971031991747828282370358243, −11.49747773618558184561399538794, −10.621369367886448464687915969238, −9.748857616581290613735297725357, −8.59007246408910956738143995071, −7.86881005308759463772286159884, −7.128816600842209552062152233402, −6.43482769569552713438684861590, −5.04334038631223034182236371513, −3.88417028365552384492681172749, −2.84613969227806618295528194425, −2.1904104256928681712548259828, −0.91968764795248348600266717295, 0.323110792757638988217364685003, 1.68386115418163296125442953751, 3.46440574420313738144682324041, 4.40770836095081749583020522477, 5.07529899855965143953268418501, 5.905717322795143740439693073929, 6.85818255172921862865652075726, 8.09414488226274692357147938135, 8.52753019763433327926018212405, 9.497700732136445694488082073848, 10.06831083738351296695070082958, 11.10564162933267255248598039372, 11.78907645776363143198981111320, 12.912525289206770089274510051860, 13.99906279059821042281506401394, 14.807499670357212686086132734990, 15.710171423995668058232459905354, 15.997690210630108259880216752890, 16.78786607235575420367061227262, 17.48001745999116931282099136551, 18.225386439435043246616691263783, 19.37796768330530144202685495844, 20.05270502659967508001897403252, 20.61378492014455097191786044060, 21.73297121473943300818281198829

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