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Properties

Label 1-4011-4011.80-r0-0-0
Degree $1$
Conductor $4011$
Sign $-0.745 + 0.666i$
Analytic cond. $18.6270$
Root an. cond. $18.6270$
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  + (0.480 + 0.876i)2-s + (−0.537 + 0.843i)4-s + (0.350 − 0.936i)5-s + (−0.997 − 0.0660i)8-s + (0.989 − 0.142i)10-s + (−0.137 + 0.990i)11-s + (−0.180 − 0.983i)13-s + (−0.421 − 0.906i)16-s + (0.0935 + 0.995i)17-s + (0.989 + 0.142i)19-s + (0.601 + 0.799i)20-s + (−0.934 + 0.355i)22-s + (0.170 + 0.985i)23-s + (−0.754 − 0.656i)25-s + (0.775 − 0.631i)26-s + ⋯
L(s)  = 1  + (0.480 + 0.876i)2-s + (−0.537 + 0.843i)4-s + (0.350 − 0.936i)5-s + (−0.997 − 0.0660i)8-s + (0.989 − 0.142i)10-s + (−0.137 + 0.990i)11-s + (−0.180 − 0.983i)13-s + (−0.421 − 0.906i)16-s + (0.0935 + 0.995i)17-s + (0.989 + 0.142i)19-s + (0.601 + 0.799i)20-s + (−0.934 + 0.355i)22-s + (0.170 + 0.985i)23-s + (−0.754 − 0.656i)25-s + (0.775 − 0.631i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.745 + 0.666i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.745 + 0.666i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4011\)    =    \(3 \cdot 7 \cdot 191\)
Sign: $-0.745 + 0.666i$
Analytic conductor: \(18.6270\)
Root analytic conductor: \(18.6270\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4011} (80, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4011,\ (0:\ ),\ -0.745 + 0.666i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5983755907 + 1.568505271i\)
\(L(\frac12)\) \(\approx\) \(0.5983755907 + 1.568505271i\)
\(L(1)\) \(\approx\) \(1.072348890 + 0.6113696786i\)
\(L(1)\) \(\approx\) \(1.072348890 + 0.6113696786i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
191 \( 1 \)
good2 \( 1 + (0.480 + 0.876i)T \)
5 \( 1 + (0.350 - 0.936i)T \)
11 \( 1 + (-0.137 + 0.990i)T \)
13 \( 1 + (-0.180 - 0.983i)T \)
17 \( 1 + (0.0935 + 0.995i)T \)
19 \( 1 + (0.989 + 0.142i)T \)
23 \( 1 + (0.170 + 0.985i)T \)
29 \( 1 + (0.0165 - 0.999i)T \)
31 \( 1 + (-0.451 + 0.892i)T \)
37 \( 1 + (0.451 + 0.892i)T \)
41 \( 1 + (-0.0825 + 0.996i)T \)
43 \( 1 + (-0.995 - 0.0990i)T \)
47 \( 1 + (-0.644 - 0.764i)T \)
53 \( 1 + (0.693 - 0.720i)T \)
59 \( 1 + (-0.889 + 0.456i)T \)
61 \( 1 + (0.660 - 0.750i)T \)
67 \( 1 + (-0.999 + 0.0110i)T \)
71 \( 1 + (0.973 - 0.229i)T \)
73 \( 1 + (-0.0715 + 0.997i)T \)
79 \( 1 + (0.874 + 0.485i)T \)
83 \( 1 + (-0.768 - 0.639i)T \)
89 \( 1 + (0.202 + 0.979i)T \)
97 \( 1 + (0.627 + 0.778i)T \)
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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

−18.31353660536852189506738475683, −18.02084169721591759469042046240, −16.79287310273284882942527946117, −16.189205774602641084489496657555, −15.292387679780463993440976665755, −14.46344684371167749045664683225, −14.06344285766538624311150736503, −13.587894377798760117312891137560, −12.75907206635013309593858285773, −11.83967958882614925534324616021, −11.34521038214436319864479795831, −10.80386473452307168826076602635, −10.06681395218513649035981228406, −9.30942816347400054908330275444, −8.8451158903264657693552426953, −7.5943894172036532387053564128, −6.84999001182402325980342198729, −6.09065338049790947054349656564, −5.41227191239584026132449853881, −4.62520512124156142779525031943, −3.67421192201415956161431829839, −3.02372297137856305910730773047, −2.42760152242438792817845375168, −1.548251126743319310924677852592, −0.42410833879423663972377982531, 1.050408445374545259887079278212, 2.02187789895467210740481972571, 3.15645448646589124922987086820, 3.866376003836147447365160989250, 4.86186786705883776537780627994, 5.19852798584979064046789875343, 5.93784684000335343588165773513, 6.735748130194651436485767248218, 7.689519598339870973790536703132, 8.06956555088894565458314563520, 8.84913058310080337154343441575, 9.80462479237219136191863821245, 10.03242406995205673407299464107, 11.48275039950435854360925569397, 12.14508035053294174842490809591, 12.80220972843423130085731297291, 13.25626501624805493973922416228, 13.88292594952237302785699254189, 14.9227167259206956068187723798, 15.235688919269582262524979127924, 15.97754783256249423522428020628, 16.73083877199952897540843753455, 17.25769122702919727849274405231, 17.855252090661021049545149068900, 18.33277446616476545143753933430

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