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Properties

Label 1-4033-4033.69-r0-0-0
Degree $1$
Conductor $4033$
Sign $0.995 + 0.0971i$
Analytic cond. $18.7291$
Root an. cond. $18.7291$
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  + 2-s + (−0.973 − 0.230i)3-s + 4-s + (0.998 + 0.0581i)5-s + (−0.973 − 0.230i)6-s + (0.893 − 0.448i)7-s + 8-s + (0.893 + 0.448i)9-s + (0.998 + 0.0581i)10-s + (−0.998 + 0.0581i)11-s + (−0.973 − 0.230i)12-s + (−0.0581 + 0.998i)13-s + (0.893 − 0.448i)14-s + (−0.957 − 0.286i)15-s + 16-s + (0.5 + 0.866i)17-s + ⋯
L(s)  = 1  + 2-s + (−0.973 − 0.230i)3-s + 4-s + (0.998 + 0.0581i)5-s + (−0.973 − 0.230i)6-s + (0.893 − 0.448i)7-s + 8-s + (0.893 + 0.448i)9-s + (0.998 + 0.0581i)10-s + (−0.998 + 0.0581i)11-s + (−0.973 − 0.230i)12-s + (−0.0581 + 0.998i)13-s + (0.893 − 0.448i)14-s + (−0.957 − 0.286i)15-s + 16-s + (0.5 + 0.866i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.995 + 0.0971i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.995 + 0.0971i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.995 + 0.0971i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (69, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ 0.995 + 0.0971i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.855214385 + 0.1877982731i\)
\(L(\frac12)\) \(\approx\) \(3.855214385 + 0.1877982731i\)
\(L(1)\) \(\approx\) \(2.053538978 + 0.02659557265i\)
\(L(1)\) \(\approx\) \(2.053538978 + 0.02659557265i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
109 \( 1 \)
good2 \( 1 + T \)
3 \( 1 + (-0.973 - 0.230i)T \)
5 \( 1 + (0.998 + 0.0581i)T \)
7 \( 1 + (0.893 - 0.448i)T \)
11 \( 1 + (-0.998 + 0.0581i)T \)
13 \( 1 + (-0.0581 + 0.998i)T \)
17 \( 1 + (0.5 + 0.866i)T \)
19 \( 1 + (0.766 - 0.642i)T \)
23 \( 1 + (0.939 + 0.342i)T \)
29 \( 1 + (-0.448 - 0.893i)T \)
31 \( 1 + (-0.116 + 0.993i)T \)
41 \( 1 + (-0.342 - 0.939i)T \)
43 \( 1 + (-0.642 + 0.766i)T \)
47 \( 1 + (-0.230 - 0.973i)T \)
53 \( 1 + (0.727 + 0.686i)T \)
59 \( 1 + (-0.286 + 0.957i)T \)
61 \( 1 + (0.802 - 0.597i)T \)
67 \( 1 + (-0.230 - 0.973i)T \)
71 \( 1 - T \)
73 \( 1 + (-0.893 - 0.448i)T \)
79 \( 1 + (-0.286 + 0.957i)T \)
83 \( 1 + (-0.396 + 0.918i)T \)
89 \( 1 + (0.549 - 0.835i)T \)
97 \( 1 + (0.116 + 0.993i)T \)
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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

−18.33073717135035714447458271025, −17.7461373509839212505731653759, −17.009294863101576972059136846098, −16.333782770268744932862548903229, −15.77193348771080363695456872540, −14.8660235368297985364202444724, −14.55503573698659965825358571900, −13.48193159229472056378917022227, −13.000374881500689685544595921373, −12.4027058014503988437069371219, −11.561586617897496371083454469961, −11.095248426465075026239053242208, −10.234779062318579638181358181096, −9.92610685199818062336138193119, −8.70933574554382057505715494163, −7.648186765887370064564804390, −7.161455349165038177554971441544, −6.03673209912342398447889756143, −5.56571708262270460411650341184, −5.118999622309009292331693722329, −4.66256878747876399135062463841, −3.335433697778696432560176641252, −2.6881964304336790634992758933, −1.72268788814141435145703229137, −0.9462062186998739445077995248, 1.1232777577965393301819247338, 1.72464886549775857522472627989, 2.44598996869779690474557575524, 3.56058262977065093788339129565, 4.5777247011179360752564405106, 5.08941537229204457174328951897, 5.57560471534910449161072905454, 6.35131927772541074377663592866, 7.14974057665817959273823245440, 7.539635918522887932510878342600, 8.6762947248180503736239648701, 9.883509950284431195648285210308, 10.44694895381571987108602739056, 11.02751631070291665813933226375, 11.64763054812041634310798124242, 12.33616292197734927351919139487, 13.27386477794818956805122005325, 13.452969380490813070026963227184, 14.24631999663597512970618514333, 14.99759481939161519649132305174, 15.70677544242056663940515197576, 16.632462175707509185495825459305, 16.978858997323322938353897530103, 17.66185966496104265579084011026, 18.34867861606721139585013678198

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