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Properties

Label 1-1001-1001.230-r0-0-0
Degree $1$
Conductor $1001$
Sign $0.252 - 0.967i$
Analytic cond. $4.64862$
Root an. cond. $4.64862$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 + 0.866i)2-s + (0.5 + 0.866i)3-s + (−0.5 + 0.866i)4-s − 5-s + (−0.5 + 0.866i)6-s − 8-s + (−0.5 + 0.866i)9-s + (−0.5 − 0.866i)10-s − 12-s + (−0.5 − 0.866i)15-s + (−0.5 − 0.866i)16-s + (−0.5 + 0.866i)17-s − 18-s + (−0.5 + 0.866i)19-s + (0.5 − 0.866i)20-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)2-s + (0.5 + 0.866i)3-s + (−0.5 + 0.866i)4-s − 5-s + (−0.5 + 0.866i)6-s − 8-s + (−0.5 + 0.866i)9-s + (−0.5 − 0.866i)10-s − 12-s + (−0.5 − 0.866i)15-s + (−0.5 − 0.866i)16-s + (−0.5 + 0.866i)17-s − 18-s + (−0.5 + 0.866i)19-s + (0.5 − 0.866i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.252 - 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.252 - 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1001\)    =    \(7 \cdot 11 \cdot 13\)
Sign: $0.252 - 0.967i$
Analytic conductor: \(4.64862\)
Root analytic conductor: \(4.64862\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1001} (230, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1001,\ (0:\ ),\ 0.252 - 0.967i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.3847701008 + 0.2972091620i\)
\(L(\frac12)\) \(\approx\) \(-0.3847701008 + 0.2972091620i\)
\(L(1)\) \(\approx\) \(0.5589965096 + 0.7545430488i\)
\(L(1)\) \(\approx\) \(0.5589965096 + 0.7545430488i\)

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 \( 1 + (0.5 + 0.866i)T \)
5 \( 1 - T \)
17 \( 1 + (-0.5 + 0.866i)T \)
19 \( 1 + (-0.5 + 0.866i)T \)
23 \( 1 + (-0.5 - 0.866i)T \)
29 \( 1 + (0.5 + 0.866i)T \)
31 \( 1 - T \)
37 \( 1 + (-0.5 - 0.866i)T \)
41 \( 1 + (-0.5 - 0.866i)T \)
43 \( 1 + (0.5 - 0.866i)T \)
47 \( 1 - T \)
53 \( 1 + T \)
59 \( 1 + (0.5 - 0.866i)T \)
61 \( 1 + (-0.5 + 0.866i)T \)
67 \( 1 + (-0.5 - 0.866i)T \)
71 \( 1 + (-0.5 + 0.866i)T \)
73 \( 1 + T \)
79 \( 1 - T \)
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

−20.861205165002296949094752001734, −19.96976953146890939401842115317, −19.71838795479256423760326022836, −18.93225492223982673514038889932, −18.21117970729745465679148796551, −17.49332236001437388968903554032, −16.07234160304758810419722222930, −15.19476648164184845625434092282, −14.63818964924273702922986985097, −13.54543649824894733695481698691, −13.194251498639298088441428842338, −12.12719996863688551592627724778, −11.658374954638561340452801503897, −10.95291628367274971403627329333, −9.69206227029299608651099861493, −8.889032432891790978080304349778, −8.0656932633085502304006143839, −7.09582858360749716696365605922, −6.27691799118010837773944049193, −5.03143874724512068608534665540, −4.12736916938757063651960786813, −3.20667081257901869537863556615, −2.4757791833943769472775417699, −1.34894199452688394657571069266, −0.16294335975297744883913519046, 2.24243599208523683523534332779, 3.53443883958498253648631722174, 3.916492545558195621862442628725, 4.766165015726737566500896455411, 5.68686306961996715247016173620, 6.78872908282959903606618759426, 7.706174099729751914440186884564, 8.51970092793073132852788622727, 8.89522484027002662694817597332, 10.26414382055439080651027026716, 10.99077172981340806350973145779, 12.15482758617427653253631268514, 12.76337344722707138172461590650, 13.88778176792972030702618577970, 14.673562345403903569597538898316, 15.07008947382549655354617316182, 15.998500403960200903982361161914, 16.38454193413067695308770988717, 17.225175479953449084178999532229, 18.32895657217366246044463687932, 19.24466215801656076131266536294, 20.06847344421473631375918678615, 20.81950810066241829426699231539, 21.64217491301543601885944366390, 22.36133266413484039131395571011

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