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Properties

Label 1-1001-1001.956-r0-0-0
Degree $1$
Conductor $1001$
Sign $-0.884 - 0.467i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 1001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.884 - 0.467i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.884 - 0.467i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1001\)    =    \(7 \cdot 11 \cdot 13\)
Sign: $-0.884 - 0.467i$
Analytic conductor: \(4.64862\)
Root analytic conductor: \(4.64862\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1001} (956, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1001,\ (0:\ ),\ -0.884 - 0.467i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.09861721817 - 0.3978009299i\)
\(L(\frac12)\) \(\approx\) \(0.09861721817 - 0.3978009299i\)
\(L(1)\) \(\approx\) \(0.5634895318 - 0.2179921483i\)
\(L(1)\) \(\approx\) \(0.5634895318 - 0.2179921483i\)

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

−22.318266956933355801983618174001, −21.59258096270558037000148103430, −20.17503623989207853043430122619, −19.11192377118719280925653701611, −18.902867749400238032858299473, −18.00633892587212210564379965863, −17.23375904089734527644763508767, −16.57500220509113326353211894868, −15.663477414488239193916203078544, −15.02629453884862007135211065601, −14.016753926076564590415306694863, −13.51037837652213511068018178404, −12.26325036242570887277879065866, −11.99116547135369588522296379983, −10.73177548538167352989453929190, −9.96156775177875195402935546326, −8.44583564755842088147599644457, −8.10395098998185681674627561770, −7.15261739901400874149486146107, −6.58733182798990517347018410108, −5.75110486773261115477529738313, −4.752548429658383173263299133301, −3.80553803403887630311667393829, −2.591098915060525840154785945440, −1.01412918056637916192672504616, 0.242254378357374080674363124434, 1.48733743769168128689325518317, 3.00029085408883309998077326413, 3.794950248207313375233336570576, 4.47184139477782554199725625508, 5.246174373672987440470917967434, 6.23397736020353859231628893666, 7.8306839289662540809160003392, 8.5141500823192092832447862060, 9.41089583458177543470661876055, 10.218436788300203450611849153220, 10.86591188452347307983776291158, 11.84879850413622543382091259761, 12.16233915772988128610320555840, 13.06238419879871105650740863850, 14.31568145050666571836364032683, 14.903761664999445892476562354020, 15.982556674268715534799085573042, 16.60942644182391594683095586137, 17.39195781712728883505084017567, 18.2982440302956850951670583221, 19.20087598179912020428675861988, 19.87639588782563756619798857214, 20.694042135455735802137094558533, 21.160652131490412223951448413251

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