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Properties

Label 1-4009-4009.1089-r0-0-0
Degree $1$
Conductor $4009$
Sign $0.523 + 0.851i$
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  + (−0.124 − 0.992i)2-s + (0.921 + 0.388i)3-s + (−0.969 + 0.246i)4-s + (0.542 + 0.840i)5-s + (0.270 − 0.962i)6-s + (0.955 + 0.294i)7-s + (0.365 + 0.930i)8-s + (0.698 + 0.715i)9-s + (0.766 − 0.642i)10-s + (−0.733 − 0.680i)11-s + (−0.988 − 0.149i)12-s + (0.980 + 0.198i)13-s + (0.173 − 0.984i)14-s + (0.173 + 0.984i)15-s + (0.878 − 0.478i)16-s + (−0.318 + 0.947i)17-s + ⋯
L(s)  = 1  + (−0.124 − 0.992i)2-s + (0.921 + 0.388i)3-s + (−0.969 + 0.246i)4-s + (0.542 + 0.840i)5-s + (0.270 − 0.962i)6-s + (0.955 + 0.294i)7-s + (0.365 + 0.930i)8-s + (0.698 + 0.715i)9-s + (0.766 − 0.642i)10-s + (−0.733 − 0.680i)11-s + (−0.988 − 0.149i)12-s + (0.980 + 0.198i)13-s + (0.173 − 0.984i)14-s + (0.173 + 0.984i)15-s + (0.878 − 0.478i)16-s + (−0.318 + 0.947i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4009 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.523 + 0.851i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4009 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.523 + 0.851i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4009\)    =    \(19 \cdot 211\)
Sign: $0.523 + 0.851i$
Analytic conductor: \(18.6177\)
Root analytic conductor: \(18.6177\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4009} (1089, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4009,\ (0:\ ),\ 0.523 + 0.851i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.066869814 + 1.155818414i\)
\(L(\frac12)\) \(\approx\) \(2.066869814 + 1.155818414i\)
\(L(1)\) \(\approx\) \(1.447710496 + 0.01910651740i\)
\(L(1)\) \(\approx\) \(1.447710496 + 0.01910651740i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad19 \( 1 \)
211 \( 1 \)
good2 \( 1 + (-0.124 - 0.992i)T \)
3 \( 1 + (0.921 + 0.388i)T \)
5 \( 1 + (0.542 + 0.840i)T \)
7 \( 1 + (0.955 + 0.294i)T \)
11 \( 1 + (-0.733 - 0.680i)T \)
13 \( 1 + (0.980 + 0.198i)T \)
17 \( 1 + (-0.318 + 0.947i)T \)
23 \( 1 + (-0.939 - 0.342i)T \)
29 \( 1 + (-0.318 - 0.947i)T \)
31 \( 1 + (0.365 + 0.930i)T \)
37 \( 1 + (-0.733 + 0.680i)T \)
41 \( 1 + (-0.318 + 0.947i)T \)
43 \( 1 + (-0.661 - 0.749i)T \)
47 \( 1 + (-0.411 + 0.911i)T \)
53 \( 1 + (-0.583 - 0.811i)T \)
59 \( 1 + (-0.318 + 0.947i)T \)
61 \( 1 + (0.766 - 0.642i)T \)
67 \( 1 + (0.921 - 0.388i)T \)
71 \( 1 + (-0.939 + 0.342i)T \)
73 \( 1 + (0.995 + 0.0995i)T \)
79 \( 1 + (-0.0249 + 0.999i)T \)
83 \( 1 + (-0.5 + 0.866i)T \)
89 \( 1 + (0.456 + 0.889i)T \)
97 \( 1 + (-0.583 + 0.811i)T \)
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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

−18.248134424205860042433139346491, −17.7259354151346739511813956534, −17.16004000517179967584883253587, −16.0709050742692023688542970835, −15.76613215327479157881614169871, −14.952200966572445286131767534848, −14.2023261218443545148565040287, −13.712236022241254278581271177, −13.19867923402660230115858944749, −12.58632242970048408149799479725, −11.63594818434488210761395477692, −10.424663053129950008062176113218, −9.84377637810229540881293029628, −9.01254820920302574729476240830, −8.57473392003947727314545162700, −7.85042447241795968228864030327, −7.4093791605280529519741210954, −6.528323990818753283244263715073, −5.61555263069107571860723703867, −4.949016519597180472112462121867, −4.29618573516298996299764780616, −3.45967935721581875711390951575, −2.1122583077915031082830502786, −1.564977792880998828109784203222, −0.574867781982293490701681415729, 1.393342441155495659808884022684, 1.956644347111751840954364852993, 2.64508173457810483942540302734, 3.39631079719446057289423068339, 4.01880419360766473630689250942, 4.916117276228204848105810632014, 5.697483720139024446396827436541, 6.633808923459187250127466300406, 7.959831825757981499267933720716, 8.2425663938348153607291680587, 8.82277012738980293914009234031, 9.82156405422297010312877377180, 10.29742888892566297165818383227, 10.99789011836825044778770759847, 11.35776730850287851160665726668, 12.47165279711064298691158227849, 13.34014545929623940689483382236, 13.80220084017831238927121530304, 14.253700314701619181146836270570, 15.053413021042814264953030490274, 15.63896363042179975582399582296, 16.64344199715778188108549941163, 17.5858769174478907678729981427, 18.12647480499467792855161117741, 18.755124626448844836406175497505

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