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Properties

Label 1-4009-4009.18-r0-0-0
Degree $1$
Conductor $4009$
Sign $0.533 + 0.845i$
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  + (0.858 − 0.512i)2-s + (−0.393 + 0.919i)3-s + (0.473 − 0.880i)4-s + (−0.393 − 0.919i)5-s + (0.134 + 0.990i)6-s + (−0.858 − 0.512i)7-s + (−0.0448 − 0.998i)8-s + (−0.691 − 0.722i)9-s + (−0.809 − 0.587i)10-s + (0.753 + 0.657i)11-s + (0.623 + 0.781i)12-s + (0.550 + 0.834i)13-s − 14-s + 15-s + (−0.550 − 0.834i)16-s + (−0.936 + 0.351i)17-s + ⋯
L(s)  = 1  + (0.858 − 0.512i)2-s + (−0.393 + 0.919i)3-s + (0.473 − 0.880i)4-s + (−0.393 − 0.919i)5-s + (0.134 + 0.990i)6-s + (−0.858 − 0.512i)7-s + (−0.0448 − 0.998i)8-s + (−0.691 − 0.722i)9-s + (−0.809 − 0.587i)10-s + (0.753 + 0.657i)11-s + (0.623 + 0.781i)12-s + (0.550 + 0.834i)13-s − 14-s + 15-s + (−0.550 − 0.834i)16-s + (−0.936 + 0.351i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4009 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.533 + 0.845i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4009 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.533 + 0.845i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4009\)    =    \(19 \cdot 211\)
Sign: $0.533 + 0.845i$
Analytic conductor: \(18.6177\)
Root analytic conductor: \(18.6177\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4009} (18, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4009,\ (0:\ ),\ 0.533 + 0.845i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9118542746 + 0.5026391087i\)
\(L(\frac12)\) \(\approx\) \(0.9118542746 + 0.5026391087i\)
\(L(1)\) \(\approx\) \(1.097020966 - 0.2118422459i\)
\(L(1)\) \(\approx\) \(1.097020966 - 0.2118422459i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad19 \( 1 \)
211 \( 1 \)
good2 \( 1 + (0.858 - 0.512i)T \)
3 \( 1 + (-0.393 + 0.919i)T \)
5 \( 1 + (-0.393 - 0.919i)T \)
7 \( 1 + (-0.858 - 0.512i)T \)
11 \( 1 + (0.753 + 0.657i)T \)
13 \( 1 + (0.550 + 0.834i)T \)
17 \( 1 + (-0.936 + 0.351i)T \)
23 \( 1 + (-0.309 - 0.951i)T \)
29 \( 1 + (-0.550 - 0.834i)T \)
31 \( 1 + (0.623 + 0.781i)T \)
37 \( 1 + (-0.753 + 0.657i)T \)
41 \( 1 + (-0.963 - 0.266i)T \)
43 \( 1 + (0.623 + 0.781i)T \)
47 \( 1 + (-0.691 + 0.722i)T \)
53 \( 1 + (-0.473 - 0.880i)T \)
59 \( 1 + (0.963 + 0.266i)T \)
61 \( 1 + (0.809 + 0.587i)T \)
67 \( 1 + (-0.222 + 0.974i)T \)
71 \( 1 + (0.809 + 0.587i)T \)
73 \( 1 + (-0.900 + 0.433i)T \)
79 \( 1 + (0.0448 - 0.998i)T \)
83 \( 1 + (-0.809 + 0.587i)T \)
89 \( 1 + (-0.995 - 0.0896i)T \)
97 \( 1 + (0.983 + 0.178i)T \)
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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

−18.35346557487149630029101146058, −17.65971688866030551945491749423, −17.00708905995567238035962200997, −16.130991963144262261671308386691, −15.62442728886900549997701423059, −15.02157769239596572624519000589, −14.08648116336675025118379624225, −13.61999364786360185761316754813, −12.99745415036622279849724841950, −12.2971480508015111526651234711, −11.55301978312370697815864395982, −11.22895205247997595403654331309, −10.35436717860792606745711570935, −9.07767307097580700958935250698, −8.37559719894814118746731116757, −7.61144258978443973361968291809, −6.87250257796168961596722173462, −6.445270775907190113034161961969, −5.8296704293073890882471564093, −5.20664928883867051875772722097, −3.85654175424616257665255236183, −3.35152658700632773453180824723, −2.641735795804824343337137048228, −1.79682341645362876057964856761, −0.25140615330428469119989513050, 0.93149079328964812892377757946, 1.85103863770895722286891943242, 3.01543438228993682924799718697, 3.95425722077613717115467465403, 4.199915093781760140873185070974, 4.76261007034325332438412605907, 5.73380375735470150110281299214, 6.53436945067709923185654508588, 6.904383668690591894039080887015, 8.43738936477830957959419906963, 9.08046183477024780467828883416, 9.814988931221600256828044950087, 10.26005031972013915188130155566, 11.28266642485562441476448874854, 11.65164006241589506719365515741, 12.423085352662858636470975730018, 12.99563098306317375202901527953, 13.753087638094260152305192845531, 14.50114050281497273078663132623, 15.2422649201502803470950190109, 15.97703460687283744303532534567, 16.23115750011642592114769157466, 17.04755741571142831559715122701, 17.67108052954548455726405906365, 18.98428616371380750246598102856

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