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Properties

Label 1-4009-4009.100-r0-0-0
Degree $1$
Conductor $4009$
Sign $-0.223 + 0.974i$
Analytic cond. $18.6177$
Root an. cond. $18.6177$
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  + (0.559 + 0.829i)2-s + (0.990 + 0.139i)3-s + (−0.374 + 0.927i)4-s + (−0.882 + 0.469i)5-s + (0.438 + 0.898i)6-s + (−0.809 + 0.587i)7-s + (−0.978 + 0.207i)8-s + (0.961 + 0.275i)9-s + (−0.882 − 0.469i)10-s + (0.669 − 0.743i)11-s + (−0.5 + 0.866i)12-s + (0.559 − 0.829i)13-s + (−0.939 − 0.342i)14-s + (−0.939 + 0.342i)15-s + (−0.719 − 0.694i)16-s + (0.961 − 0.275i)17-s + ⋯
L(s)  = 1  + (0.559 + 0.829i)2-s + (0.990 + 0.139i)3-s + (−0.374 + 0.927i)4-s + (−0.882 + 0.469i)5-s + (0.438 + 0.898i)6-s + (−0.809 + 0.587i)7-s + (−0.978 + 0.207i)8-s + (0.961 + 0.275i)9-s + (−0.882 − 0.469i)10-s + (0.669 − 0.743i)11-s + (−0.5 + 0.866i)12-s + (0.559 − 0.829i)13-s + (−0.939 − 0.342i)14-s + (−0.939 + 0.342i)15-s + (−0.719 − 0.694i)16-s + (0.961 − 0.275i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4009 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.223 + 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4009 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.223 + 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4009\)    =    \(19 \cdot 211\)
Sign: $-0.223 + 0.974i$
Analytic conductor: \(18.6177\)
Root analytic conductor: \(18.6177\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4009} (100, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4009,\ (0:\ ),\ -0.223 + 0.974i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.759317299 + 2.209528736i\)
\(L(\frac12)\) \(\approx\) \(1.759317299 + 2.209528736i\)
\(L(1)\) \(\approx\) \(1.339395338 + 0.9686234556i\)
\(L(1)\) \(\approx\) \(1.339395338 + 0.9686234556i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad19 \( 1 \)
211 \( 1 \)
good2 \( 1 + (0.559 + 0.829i)T \)
3 \( 1 + (0.990 + 0.139i)T \)
5 \( 1 + (-0.882 + 0.469i)T \)
7 \( 1 + (-0.809 + 0.587i)T \)
11 \( 1 + (0.669 - 0.743i)T \)
13 \( 1 + (0.559 - 0.829i)T \)
17 \( 1 + (0.961 - 0.275i)T \)
23 \( 1 + (-0.719 + 0.694i)T \)
29 \( 1 + (0.559 - 0.829i)T \)
31 \( 1 + T \)
37 \( 1 + (-0.978 + 0.207i)T \)
41 \( 1 + (0.848 + 0.529i)T \)
43 \( 1 + (-0.939 - 0.342i)T \)
47 \( 1 + (0.438 + 0.898i)T \)
53 \( 1 + (0.0348 + 0.999i)T \)
59 \( 1 + (0.0348 - 0.999i)T \)
61 \( 1 + (0.848 - 0.529i)T \)
67 \( 1 + (0.766 + 0.642i)T \)
71 \( 1 + (0.990 + 0.139i)T \)
73 \( 1 + (0.766 - 0.642i)T \)
79 \( 1 + (0.0348 - 0.999i)T \)
83 \( 1 + (0.309 + 0.951i)T \)
89 \( 1 + (0.559 - 0.829i)T \)
97 \( 1 + (-0.374 - 0.927i)T \)
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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

−18.788850083120935857099410785557, −17.85340055910018127663897908287, −16.73340074693757855603407925216, −16.07245930902705077186195368810, −15.44964651815833884747598112995, −14.63558472108352775230430246956, −14.12319252385093855795485044560, −13.50968817837106290009418728794, −12.67608665033063568258235845438, −12.273512872865946463976825426710, −11.7123118499821795124695563877, −10.58310643793660565228477989548, −10.02357733421421036654151382233, −9.326248365203509358936524158958, −8.6885034189929881870945861024, −7.96484479064150377724178667589, −6.836699912175311229499664203971, −6.60156999012581625344575457022, −5.227253897366733870229974893089, −4.28729906928348833584981347861, −3.872303543752109544579899510627, −3.411761433846006440684714077352, −2.41651642360871422311028422259, −1.48677546827929535757475566685, −0.814260208162545716613859506788, 0.82961120351125817676572757917, 2.452211390256984043189940431245, 3.26045839289759063973363853022, 3.4637620735273079886081406424, 4.22338008953150151384189241907, 5.26553838106387123012656091058, 6.15946494669622837229301905816, 6.66436394148044682510661047222, 7.63794498390962613626355612936, 8.11996980729334033190217458452, 8.64213345158666173687357854634, 9.50764513504120980879832405591, 10.16844317682229450828135878854, 11.31094151493737082408779893937, 12.0866565003834631311425473426, 12.57398694783672184719557533882, 13.51411598719863700313156641328, 13.98128191813729481502675478946, 14.62140716998204229237701023608, 15.45205120905397733206738392442, 15.72071150927684669484576240085, 16.19655039231851885608687914566, 17.12381982320581588185668443274, 18.095341736154352152742254864287, 18.84089407813683816263830486258

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