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Properties

Label 1-4003-4003.1618-r0-0-0
Degree $1$
Conductor $4003$
Sign $0.271 + 0.962i$
Analytic cond. $18.5898$
Root an. cond. $18.5898$
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  + (0.126 − 0.992i)2-s + (−0.674 + 0.738i)3-s + (−0.968 − 0.250i)4-s + (0.530 − 0.847i)5-s + (0.647 + 0.762i)6-s + (−0.891 + 0.452i)7-s + (−0.370 + 0.928i)8-s + (−0.0901 − 0.995i)9-s + (−0.773 − 0.633i)10-s + (−0.856 + 0.515i)11-s + (0.837 − 0.546i)12-s + (−0.725 − 0.687i)13-s + (0.336 + 0.941i)14-s + (0.267 + 0.963i)15-s + (0.874 + 0.484i)16-s + (−0.947 − 0.319i)17-s + ⋯
L(s)  = 1  + (0.126 − 0.992i)2-s + (−0.674 + 0.738i)3-s + (−0.968 − 0.250i)4-s + (0.530 − 0.847i)5-s + (0.647 + 0.762i)6-s + (−0.891 + 0.452i)7-s + (−0.370 + 0.928i)8-s + (−0.0901 − 0.995i)9-s + (−0.773 − 0.633i)10-s + (−0.856 + 0.515i)11-s + (0.837 − 0.546i)12-s + (−0.725 − 0.687i)13-s + (0.336 + 0.941i)14-s + (0.267 + 0.963i)15-s + (0.874 + 0.484i)16-s + (−0.947 − 0.319i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4003 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.271 + 0.962i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4003 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.271 + 0.962i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4003\)
Sign: $0.271 + 0.962i$
Analytic conductor: \(18.5898\)
Root analytic conductor: \(18.5898\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4003} (1618, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4003,\ (0:\ ),\ 0.271 + 0.962i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.08723705865 + 0.06601441159i\)
\(L(\frac12)\) \(\approx\) \(0.08723705865 + 0.06601441159i\)
\(L(1)\) \(\approx\) \(0.5612701465 - 0.2656761386i\)
\(L(1)\) \(\approx\) \(0.5612701465 - 0.2656761386i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad4003 \( 1 \)
good2 \( 1 + (0.126 - 0.992i)T \)
3 \( 1 + (-0.674 + 0.738i)T \)
5 \( 1 + (0.530 - 0.847i)T \)
7 \( 1 + (-0.891 + 0.452i)T \)
11 \( 1 + (-0.856 + 0.515i)T \)
13 \( 1 + (-0.725 - 0.687i)T \)
17 \( 1 + (-0.947 - 0.319i)T \)
19 \( 1 + (0.197 + 0.980i)T \)
23 \( 1 + (0.874 + 0.484i)T \)
29 \( 1 + (0.197 - 0.980i)T \)
31 \( 1 + (0.700 - 0.713i)T \)
37 \( 1 + (0.197 - 0.980i)T \)
41 \( 1 + (-0.856 - 0.515i)T \)
43 \( 1 + (-0.302 + 0.953i)T \)
47 \( 1 + (0.874 - 0.484i)T \)
53 \( 1 + (0.874 - 0.484i)T \)
59 \( 1 + (-0.619 - 0.785i)T \)
61 \( 1 + (-0.983 - 0.179i)T \)
67 \( 1 + (0.336 - 0.941i)T \)
71 \( 1 + (-0.674 + 0.738i)T \)
73 \( 1 + (-0.561 - 0.827i)T \)
79 \( 1 + (-0.983 + 0.179i)T \)
83 \( 1 + (0.126 - 0.992i)T \)
89 \( 1 + (0.336 + 0.941i)T \)
97 \( 1 + (0.336 - 0.941i)T \)
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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

−18.2108911559896606389488198579, −17.582350634328411693991055589529, −16.96355706250113528373025217697, −16.49435095177460392162411630421, −15.62900820769208313165561529980, −15.08015242305912114905305374630, −14.009111686735381702298308168749, −13.60931939179566449738158879669, −13.14003477787175862546668488524, −12.437384657120647873581757593, −11.49720145776492213595954150412, −10.55041184260231202530825287783, −10.2304859896115694522224691412, −9.14562674735217374951337378923, −8.515513279701354057474260815350, −7.34005570349998723890946570964, −7.005396950701971284310553427310, −6.549925586989503021908830002686, −5.86166805191899222239166957202, −5.044212680402535503107105616045, −4.39730952959255765261765223504, −3.03394755114771560068329930657, −2.642495721429847707234027395453, −1.24495297129243233593134882301, −0.04696302610376783973199507198, 0.743233006966584873185678204781, 2.03315854116602742351233052689, 2.701873893569223061459395315893, 3.56467880880504786182049600264, 4.49938852757232092775876866028, 5.016729230956806570243158562681, 5.661418420757500719712310962712, 6.226571221054541699049278839915, 7.533916888334599046055791129133, 8.57144403328410050968393772026, 9.24670861381274912407548665962, 9.87984735450829999030914155317, 10.12898678110689495728241147950, 10.98429606980021795344153976519, 11.98159394237198385689135988136, 12.26031203782038373432156419000, 13.12034799973863800756959210515, 13.364549633255164983749081796085, 14.55944767418697862169198308397, 15.417530612735303502373237063140, 15.76036220558977739944411021803, 16.84103828682639595771316777972, 17.23946497946584384937181811313, 17.97484464887150104424814991552, 18.53172742996496132736065259357

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