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Properties

Label 1-4009-4009.409-r0-0-0
Degree $1$
Conductor $4009$
Sign $0.685 + 0.728i$
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  + (0.683 + 0.729i)2-s + (−0.0846 + 0.996i)3-s + (−0.0647 + 0.997i)4-s + (−0.705 − 0.708i)5-s + (−0.784 + 0.619i)6-s + (−0.992 + 0.119i)7-s + (−0.772 + 0.635i)8-s + (−0.985 − 0.168i)9-s + (0.0348 − 0.999i)10-s + (−0.873 + 0.486i)11-s + (−0.988 − 0.149i)12-s + (−0.976 + 0.217i)13-s + (−0.766 − 0.642i)14-s + (0.766 − 0.642i)15-s + (−0.991 − 0.129i)16-s + (−0.814 + 0.579i)17-s + ⋯
L(s)  = 1  + (0.683 + 0.729i)2-s + (−0.0846 + 0.996i)3-s + (−0.0647 + 0.997i)4-s + (−0.705 − 0.708i)5-s + (−0.784 + 0.619i)6-s + (−0.992 + 0.119i)7-s + (−0.772 + 0.635i)8-s + (−0.985 − 0.168i)9-s + (0.0348 − 0.999i)10-s + (−0.873 + 0.486i)11-s + (−0.988 − 0.149i)12-s + (−0.976 + 0.217i)13-s + (−0.766 − 0.642i)14-s + (0.766 − 0.642i)15-s + (−0.991 − 0.129i)16-s + (−0.814 + 0.579i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4009 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.685 + 0.728i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4009 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.685 + 0.728i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4009\)    =    \(19 \cdot 211\)
Sign: $0.685 + 0.728i$
Analytic conductor: \(18.6177\)
Root analytic conductor: \(18.6177\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4009} (409, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4009,\ (0:\ ),\ 0.685 + 0.728i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1411597132 + 0.06101500841i\)
\(L(\frac12)\) \(\approx\) \(0.1411597132 + 0.06101500841i\)
\(L(1)\) \(\approx\) \(0.4723369570 + 0.5139054396i\)
\(L(1)\) \(\approx\) \(0.4723369570 + 0.5139054396i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad19 \( 1 \)
211 \( 1 \)
good2 \( 1 + (0.683 + 0.729i)T \)
3 \( 1 + (-0.0846 + 0.996i)T \)
5 \( 1 + (-0.705 - 0.708i)T \)
7 \( 1 + (-0.992 + 0.119i)T \)
11 \( 1 + (-0.873 + 0.486i)T \)
13 \( 1 + (-0.976 + 0.217i)T \)
17 \( 1 + (-0.814 + 0.579i)T \)
23 \( 1 + (0.241 - 0.970i)T \)
29 \( 1 + (-0.299 + 0.954i)T \)
31 \( 1 + (-0.988 - 0.149i)T \)
37 \( 1 + (-0.858 + 0.512i)T \)
41 \( 1 + (-0.917 + 0.397i)T \)
43 \( 1 + (0.980 - 0.198i)T \)
47 \( 1 + (-0.863 - 0.504i)T \)
53 \( 1 + (0.993 + 0.109i)T \)
59 \( 1 + (-0.803 - 0.595i)T \)
61 \( 1 + (-0.848 - 0.529i)T \)
67 \( 1 + (-0.124 + 0.992i)T \)
71 \( 1 + (0.374 - 0.927i)T \)
73 \( 1 + (-0.583 + 0.811i)T \)
79 \( 1 + (0.183 + 0.983i)T \)
83 \( 1 + (-0.978 + 0.207i)T \)
89 \( 1 + (0.778 + 0.627i)T \)
97 \( 1 + (0.212 + 0.977i)T \)
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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

−18.55058531126593163126980365093, −18.09328589032002724427847788998, −17.18733611707693330912025993869, −16.12974010458851525211663022559, −15.53299737842782002771520428318, −14.89144701375377845649602161899, −14.0284486995623485606670798041, −13.45957231592062813452355111624, −12.95173282387777272899751787418, −12.21613838124792957107618914457, −11.68719839378203301934742764618, −10.94886600108923906486394593239, −10.41981100292544595160105475523, −9.50642474094065704178153035679, −8.72683627196612234552655810912, −7.45167294893697664216810678609, −7.29118168306497923606083339520, −6.348984051402820043065696939806, −5.712877210212960009399534264541, −4.91846970224712109426451931252, −3.84265615421947101737782218, −3.05188628517525981393844500278, −2.67667577328141332209563292779, −1.83602741134544142781546482635, −0.41827500383681052946417425378, 0.07297797571100006180130884070, 2.197622831741618430017775759297, 3.0445516196063598271963434080, 3.73771827726298467700076122092, 4.46206803815636421504306698667, 5.02035281935334575251950603285, 5.58352643482199978114999247494, 6.62114603075518647215613932971, 7.21890707448251748126024184320, 8.15816200773347613575824567367, 8.809967867280622831518726507905, 9.3631805996559588701203642447, 10.263189558752460036765651166028, 11.032357009254080881296897988863, 11.94523799303109957104699450421, 12.591111028620028510375143021586, 12.93684615736618320391128162024, 13.881518894043232312520828934186, 14.86386002844825340382857780383, 15.22916644526168480841353651762, 15.766876038047848571125442522105, 16.51385501215991583640923542705, 16.73571881735505396668784175739, 17.53062818772483300592865418477, 18.47168008227784090466649396296

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