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Properties

Label 1-4009-4009.2450-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: $\chi_{4009} (2450, \cdot )$
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

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

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