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Properties

Label 1-1001-1001.153-r0-0-0
Degree $1$
Conductor $1001$
Sign $0.859 - 0.511i$
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 + (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

\[\begin{aligned}\Lambda(s)=\mathstrut & 1001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.859 - 0.511i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.859 - 0.511i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1001\)    =    \(7 \cdot 11 \cdot 13\)
Sign: $0.859 - 0.511i$
Analytic conductor: \(4.64862\)
Root analytic conductor: \(4.64862\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1001} (153, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1001,\ (0:\ ),\ 0.859 - 0.511i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.503279484 - 0.4131512632i\)
\(L(\frac12)\) \(\approx\) \(1.503279484 - 0.4131512632i\)
\(L(1)\) \(\approx\) \(1.123613123 - 0.03807746655i\)
\(L(1)\) \(\approx\) \(1.123613123 - 0.03807746655i\)

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

−21.6737211410974545182382814967, −20.91910418976712528335775868791, −20.18172808374121405753417248222, −19.65532854526879898887551271725, −18.636545403406012538622436650192, −17.835629593015038740994752792601, −17.11646514420346863862659773669, −16.43316638538267162293005296223, −15.467292210061164780136279783057, −14.4386141228266429377695869177, −13.68817183919431020071014071095, −13.082397311251059458872581005104, −12.031865719962846841364942745057, −10.96460375775868589095893199349, −10.35311358242352364718968071282, −9.77621212738408571263721940741, −8.84802841577267892986494907014, −8.46712919391560025426057688744, −7.19536557356126045062208669383, −5.96126596987801200132318277827, −4.80845323427005505191126117878, −4.15702498249420866042773582475, −2.8947137699255779109737288784, −2.424298699746452759175366638476, −1.228187608889636732452043781629, 0.83174112101978445213597303648, 1.81777192665794341067652060950, 2.72596327789202210958127842522, 4.21160579188988254844813216283, 5.5176697183031386429496032588, 6.057044700610404216602529425290, 6.93644485386004093313486768283, 7.68325515829740525198174619622, 8.49835234399785666824541152597, 9.43263918037178829694436434635, 9.83265570510468055333207536003, 11.057708196270689514629390510258, 12.19178700824992614724190603964, 13.18413678342874374548592970510, 14.02382509598670817254048843142, 14.10597381526487204064836566427, 15.315167047085665042170639409, 16.08779334226121474327294889163, 17.17828548928198010227361898107, 17.6766613703140535319841511167, 18.325791495255963543239675003968, 18.98284880181413544272126946399, 19.86732150297407111098375557049, 20.60865722115205400076044679280, 21.57527246462612679790513897237

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