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Properties

Label 1-1001-1001.17-r0-0-0
Degree $1$
Conductor $1001$
Sign $0.836 + 0.548i$
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  + (−0.809 + 0.587i)2-s + (0.978 + 0.207i)3-s + (0.309 − 0.951i)4-s + (0.913 − 0.406i)5-s + (−0.913 + 0.406i)6-s + (0.309 + 0.951i)8-s + (0.913 + 0.406i)9-s + (−0.5 + 0.866i)10-s + (0.5 − 0.866i)12-s + (0.978 − 0.207i)15-s + (−0.809 − 0.587i)16-s + (−0.809 − 0.587i)17-s + (−0.978 + 0.207i)18-s + (−0.669 + 0.743i)19-s + (−0.104 − 0.994i)20-s + ⋯
L(s)  = 1  + (−0.809 + 0.587i)2-s + (0.978 + 0.207i)3-s + (0.309 − 0.951i)4-s + (0.913 − 0.406i)5-s + (−0.913 + 0.406i)6-s + (0.309 + 0.951i)8-s + (0.913 + 0.406i)9-s + (−0.5 + 0.866i)10-s + (0.5 − 0.866i)12-s + (0.978 − 0.207i)15-s + (−0.809 − 0.587i)16-s + (−0.809 − 0.587i)17-s + (−0.978 + 0.207i)18-s + (−0.669 + 0.743i)19-s + (−0.104 − 0.994i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.836 + 0.548i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.836 + 0.548i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1001\)    =    \(7 \cdot 11 \cdot 13\)
Sign: $0.836 + 0.548i$
Analytic conductor: \(4.64862\)
Root analytic conductor: \(4.64862\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1001} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1001,\ (0:\ ),\ 0.836 + 0.548i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.724778942 + 0.5151082305i\)
\(L(\frac12)\) \(\approx\) \(1.724778942 + 0.5151082305i\)
\(L(1)\) \(\approx\) \(1.206090917 + 0.2773717875i\)
\(L(1)\) \(\approx\) \(1.206090917 + 0.2773717875i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
11 \( 1 \)
13 \( 1 \)
good2 \( 1 + (-0.809 + 0.587i)T \)
3 \( 1 + (0.978 + 0.207i)T \)
5 \( 1 + (0.913 - 0.406i)T \)
17 \( 1 + (-0.809 - 0.587i)T \)
19 \( 1 + (-0.669 + 0.743i)T \)
23 \( 1 + T \)
29 \( 1 + (0.978 - 0.207i)T \)
31 \( 1 + (0.913 + 0.406i)T \)
37 \( 1 + (-0.309 + 0.951i)T \)
41 \( 1 + (-0.669 + 0.743i)T \)
43 \( 1 + (0.5 - 0.866i)T \)
47 \( 1 + (-0.978 - 0.207i)T \)
53 \( 1 + (0.913 + 0.406i)T \)
59 \( 1 + (0.309 - 0.951i)T \)
61 \( 1 + (-0.104 + 0.994i)T \)
67 \( 1 + (0.5 - 0.866i)T \)
71 \( 1 + (-0.913 + 0.406i)T \)
73 \( 1 + (0.978 - 0.207i)T \)
79 \( 1 + (0.104 + 0.994i)T \)
83 \( 1 + (0.809 + 0.587i)T \)
89 \( 1 + T \)
97 \( 1 + (-0.104 - 0.994i)T \)
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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

−21.32572088264998122548309830147, −20.84079455215413208696416139648, −19.82219675039156484678482338349, −19.29716251404438229394242914778, −18.61030509287363831098283009383, −17.64072327653702047263555888606, −17.38561649216414603169592532198, −16.111425559382111859847968758508, −15.2332731206230305919975115672, −14.44707337234839990612292165483, −13.247923637693558660588568959235, −13.16323501140745920829380827157, −11.98453569286043697350172328208, −10.80397087514544864205962934861, −10.3183843600374097469358299947, −9.295187511567745639891698797947, −8.86066310204389137078532744603, −7.98406731180993298306921187653, −6.90302439398747865020405789540, −6.44069890740141002903074223571, −4.747989285882324640382712408396, −3.65482889271619623056623747631, −2.65127369350954548528596520950, −2.16198503885963465320319524679, −1.093472647199202315285688828079, 1.133035429989056558735132672094, 2.0630086480424858746032208912, 2.90633781498197506161149535441, 4.48744196884178408126873500976, 5.18236763113352755045873785848, 6.413247593362243697446597007355, 7.008222255266125147082134447033, 8.309293036073160460648649643749, 8.60468672909853720503390126471, 9.55062174836288687728652166735, 10.08633013063971079629064819097, 10.89538627238464996054800482321, 12.2404077361292398919142969390, 13.42758723465834582634250218054, 13.83038889788560841836985113486, 14.77011321550766421199986218538, 15.42315748428249106601365030812, 16.27794860516791850561707960399, 17.005154837993943543746477648447, 17.80247607363811787576270270590, 18.58596862027865261893379730685, 19.32484067614467752550836590144, 20.10730780510031242806785034576, 20.82404592320729354486276321700, 21.39260808581550995531985828613

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