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Properties

Label 1-4033-4033.392-r0-0-0
Degree $1$
Conductor $4033$
Sign $-0.0379 - 0.999i$
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  + (−0.5 + 0.866i)2-s + (0.0581 + 0.998i)3-s + (−0.5 − 0.866i)4-s + (−0.918 + 0.396i)5-s + (−0.893 − 0.448i)6-s + (−0.993 − 0.116i)7-s + 8-s + (−0.993 + 0.116i)9-s + (0.116 − 0.993i)10-s + (−0.116 − 0.993i)11-s + (0.835 − 0.549i)12-s + (0.396 + 0.918i)13-s + (0.597 − 0.802i)14-s + (−0.448 − 0.893i)15-s + (−0.5 + 0.866i)16-s + (0.5 + 0.866i)17-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)2-s + (0.0581 + 0.998i)3-s + (−0.5 − 0.866i)4-s + (−0.918 + 0.396i)5-s + (−0.893 − 0.448i)6-s + (−0.993 − 0.116i)7-s + 8-s + (−0.993 + 0.116i)9-s + (0.116 − 0.993i)10-s + (−0.116 − 0.993i)11-s + (0.835 − 0.549i)12-s + (0.396 + 0.918i)13-s + (0.597 − 0.802i)14-s + (−0.448 − 0.893i)15-s + (−0.5 + 0.866i)16-s + (0.5 + 0.866i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0379 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0379 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $-0.0379 - 0.999i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (392, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ -0.0379 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.2863336763 + 0.2974060779i\)
\(L(\frac12)\) \(\approx\) \(-0.2863336763 + 0.2974060779i\)
\(L(1)\) \(\approx\) \(0.3466995871 + 0.4202721240i\)
\(L(1)\) \(\approx\) \(0.3466995871 + 0.4202721240i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
109 \( 1 \)
good2 \( 1 + (-0.5 + 0.866i)T \)
3 \( 1 + (0.0581 + 0.998i)T \)
5 \( 1 + (-0.918 + 0.396i)T \)
7 \( 1 + (-0.993 - 0.116i)T \)
11 \( 1 + (-0.116 - 0.993i)T \)
13 \( 1 + (0.396 + 0.918i)T \)
17 \( 1 + (0.5 + 0.866i)T \)
19 \( 1 + (-0.939 + 0.342i)T \)
23 \( 1 + (-0.766 - 0.642i)T \)
29 \( 1 + (0.802 + 0.597i)T \)
31 \( 1 + (-0.230 + 0.973i)T \)
41 \( 1 + (-0.342 + 0.939i)T \)
43 \( 1 + (0.984 - 0.173i)T \)
47 \( 1 + (0.448 + 0.893i)T \)
53 \( 1 + (-0.448 + 0.893i)T \)
59 \( 1 + (0.893 - 0.448i)T \)
61 \( 1 + (0.230 + 0.973i)T \)
67 \( 1 + (0.549 - 0.835i)T \)
71 \( 1 + (0.5 + 0.866i)T \)
73 \( 1 + (-0.597 - 0.802i)T \)
79 \( 1 + (-0.0581 + 0.998i)T \)
83 \( 1 + (-0.973 + 0.230i)T \)
89 \( 1 + (-0.116 + 0.993i)T \)
97 \( 1 + (0.230 + 0.973i)T \)
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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

−18.17748146982718256298768885905, −17.40910397183176330113497590199, −16.9105925539755423700912379417, −15.90903488843873543531227602578, −15.48880074008089971359595390096, −14.38778970963615113467843222588, −13.43970872435738170840291799306, −12.908476732016639972065130069, −12.49758153738456846745113087981, −11.8517037263503227795590217692, −11.32862677238524501623412852469, −10.30464510790082287016184467632, −9.704398325727342965432047483000, −8.82608382885118731578447613541, −8.28026623429257025960031492878, −7.47021891775549490568404125088, −7.145140866146248202226157241910, −6.0469278838811553243670895031, −5.109997397578294135639471702751, −4.09402592930754653095483464041, −3.42220274143403882083275068407, −2.63400244574680582229237008177, −1.95695959517643294328527139920, −0.76146756960529435312977686198, −0.22708420288664059015917384733, 1.020214665312945529600980184092, 2.53673874299202345820056900928, 3.52771339011958139787266712486, 4.00390448719103867272290419342, 4.70794460778807488550176717770, 5.8833773348509205408390091468, 6.2703382447649392670827814492, 6.98081476584974280447755215806, 8.09739291639319364563397835816, 8.49462818236654790647405337504, 9.11015194593466862752781765251, 10.01183443489690952150823998230, 10.632661580193966507442930069565, 10.97958513973038272849948472290, 12.0522301685462880210587672279, 12.878253326444952416585329869286, 14.094387873207234012904597607079, 14.240452964606394339862165274700, 15.09380846803382602407058995183, 15.84309859691171827056128939486, 16.19806643172281151842001019064, 16.60213515078776442817099337386, 17.32433090898857343260774849945, 18.452903011704149133009375077615, 18.9590399514585482479214487006

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