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Properties

Label 1-1001-1001.901-r0-0-0
Degree $1$
Conductor $1001$
Sign $0.113 + 0.993i$
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  + (−0.5 + 0.866i)2-s − 3-s + (−0.5 − 0.866i)4-s + (−0.5 − 0.866i)5-s + (0.5 − 0.866i)6-s + 8-s + 9-s + 10-s + (0.5 + 0.866i)12-s + (0.5 + 0.866i)15-s + (−0.5 + 0.866i)16-s + (−0.5 − 0.866i)17-s + (−0.5 + 0.866i)18-s − 19-s + (−0.5 + 0.866i)20-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)2-s − 3-s + (−0.5 − 0.866i)4-s + (−0.5 − 0.866i)5-s + (0.5 − 0.866i)6-s + 8-s + 9-s + 10-s + (0.5 + 0.866i)12-s + (0.5 + 0.866i)15-s + (−0.5 + 0.866i)16-s + (−0.5 − 0.866i)17-s + (−0.5 + 0.866i)18-s − 19-s + (−0.5 + 0.866i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.113 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.113 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1001\)    =    \(7 \cdot 11 \cdot 13\)
Sign: $0.113 + 0.993i$
Analytic conductor: \(4.64862\)
Root analytic conductor: \(4.64862\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1001} (901, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1001,\ (0:\ ),\ 0.113 + 0.993i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3101297164 + 0.2766675687i\)
\(L(\frac12)\) \(\approx\) \(0.3101297164 + 0.2766675687i\)
\(L(1)\) \(\approx\) \(0.4713682522 + 0.1168838833i\)
\(L(1)\) \(\approx\) \(0.4713682522 + 0.1168838833i\)

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 \( 1 - T \)
5 \( 1 + (-0.5 - 0.866i)T \)
17 \( 1 + (-0.5 - 0.866i)T \)
19 \( 1 - T \)
23 \( 1 + (-0.5 + 0.866i)T \)
29 \( 1 + (0.5 + 0.866i)T \)
31 \( 1 + (-0.5 + 0.866i)T \)
37 \( 1 + (0.5 - 0.866i)T \)
41 \( 1 + (0.5 + 0.866i)T \)
43 \( 1 + (0.5 - 0.866i)T \)
47 \( 1 + (-0.5 - 0.866i)T \)
53 \( 1 + (-0.5 + 0.866i)T \)
59 \( 1 + (-0.5 - 0.866i)T \)
61 \( 1 + T \)
67 \( 1 - T \)
71 \( 1 + (0.5 - 0.866i)T \)
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 \( 1 + (-0.5 + 0.866i)T \)
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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

−21.52409258518110435653204739091, −20.80258736509189415087126202112, −19.65393061708941344705860722747, −19.08085690539646420144734767808, −18.40944227035578238427728859651, −17.670349521727603086805674819, −17.04134336658617958333410938901, −16.164498656131387252649317661717, −15.29735747819321158197103373248, −14.329312712927029100167374953061, −13.10832719497066617919169731678, −12.53758023039475969772660386776, −11.60878520101236615701324508916, −11.07136565121951179998477977881, −10.41315851721412830256372885507, −9.7504535362563333501024964634, −8.46117384138943318566122501770, −7.71076755956890698801879606115, −6.69937899667177243452546242716, −5.98586988696731931431120550332, −4.399494628791969713598054962239, −4.08335311746112999501515921631, −2.742490257872930938427412940848, −1.81054237442315014363630954122, −0.36621916874900506272839100332, 0.76379218205269310586329319273, 1.82520592912738855086932058207, 3.87434139269327610395652398906, 4.753555929153386743345097237037, 5.3231846887392909341999436583, 6.277964492260114832636988883702, 7.12057949761714994686058730529, 7.884220623580766546696717096380, 8.88949378342201268927227599148, 9.54812653766253565390807485667, 10.61461561339160024792036891377, 11.305868876603608008400753048532, 12.313058051966733984524450901049, 13.03051470010562948204091471206, 13.9674261873820970298663730769, 15.11120085878073114872136200248, 15.86828112240136692676337851349, 16.315588540295000243598740349542, 17.04645205821174520142117924532, 17.789711615885091025517772015606, 18.3859700992036177930814594918, 19.43913167967740077075255833352, 19.98912061612869246769819006296, 21.172041328353629020151711032850, 22.0111299046494014155207060499

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