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Properties

Label 1-4033-4033.2263-r1-0-0
Degree $1$
Conductor $4033$
Sign $-0.843 + 0.537i$
Analytic cond. $433.406$
Root an. cond. $433.406$
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  + (−0.342 − 0.939i)2-s + (0.286 + 0.957i)3-s + (−0.766 + 0.642i)4-s + (−0.998 − 0.0581i)5-s + (0.802 − 0.597i)6-s + (0.893 − 0.448i)7-s + (0.866 + 0.5i)8-s + (−0.835 + 0.549i)9-s + (0.286 + 0.957i)10-s + (−0.286 + 0.957i)11-s + (−0.835 − 0.549i)12-s + (−0.448 − 0.893i)13-s + (−0.727 − 0.686i)14-s + (−0.230 − 0.973i)15-s + (0.173 − 0.984i)16-s + (−0.984 − 0.173i)17-s + ⋯
L(s)  = 1  + (−0.342 − 0.939i)2-s + (0.286 + 0.957i)3-s + (−0.766 + 0.642i)4-s + (−0.998 − 0.0581i)5-s + (0.802 − 0.597i)6-s + (0.893 − 0.448i)7-s + (0.866 + 0.5i)8-s + (−0.835 + 0.549i)9-s + (0.286 + 0.957i)10-s + (−0.286 + 0.957i)11-s + (−0.835 − 0.549i)12-s + (−0.448 − 0.893i)13-s + (−0.727 − 0.686i)14-s + (−0.230 − 0.973i)15-s + (0.173 − 0.984i)16-s + (−0.984 − 0.173i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.843 + 0.537i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.843 + 0.537i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $-0.843 + 0.537i$
Analytic conductor: \(433.406\)
Root analytic conductor: \(433.406\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (2263, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (1:\ ),\ -0.843 + 0.537i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1312749526 - 0.4500092862i\)
\(L(\frac12)\) \(\approx\) \(-0.1312749526 - 0.4500092862i\)
\(L(1)\) \(\approx\) \(0.6748191049 - 0.2155905582i\)
\(L(1)\) \(\approx\) \(0.6748191049 - 0.2155905582i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
109 \( 1 \)
good2 \( 1 + (-0.342 - 0.939i)T \)
3 \( 1 + (0.286 + 0.957i)T \)
5 \( 1 + (-0.998 - 0.0581i)T \)
7 \( 1 + (0.893 - 0.448i)T \)
11 \( 1 + (-0.286 + 0.957i)T \)
13 \( 1 + (-0.448 - 0.893i)T \)
17 \( 1 + (-0.984 - 0.173i)T \)
19 \( 1 + (0.642 - 0.766i)T \)
23 \( 1 + (-0.642 - 0.766i)T \)
29 \( 1 + (0.230 - 0.973i)T \)
31 \( 1 + (0.448 - 0.893i)T \)
41 \( 1 + (-0.5 - 0.866i)T \)
43 \( 1 + (-0.342 - 0.939i)T \)
47 \( 1 + (0.993 + 0.116i)T \)
53 \( 1 + (0.0581 - 0.998i)T \)
59 \( 1 + (-0.230 - 0.973i)T \)
61 \( 1 + (0.116 + 0.993i)T \)
67 \( 1 + (0.893 + 0.448i)T \)
71 \( 1 + (-0.939 - 0.342i)T \)
73 \( 1 + (0.286 - 0.957i)T \)
79 \( 1 + (0.549 + 0.835i)T \)
83 \( 1 + (-0.686 + 0.727i)T \)
89 \( 1 + (0.802 - 0.597i)T \)
97 \( 1 + (0.549 - 0.835i)T \)
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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

−18.523891540125146402397065964022, −18.18563108738568100596920079303, −17.41997636876825177556937803940, −16.64172409503138344727891266186, −15.95325195099678150847335491820, −15.34390512866794576146571698470, −14.5366329177855717036058487321, −14.12755496234984654126365954459, −13.51055743659166405929175676170, −12.549754266923881861187620701674, −11.81648638065355168118102790786, −11.30702192214489874778719140343, −10.46887473327568435269632247711, −9.21261058140657907450415222745, −8.71956430189665395773102431933, −8.0887766433573087708643172285, −7.66728589196256628685284700346, −6.91210337611564959436196409697, −6.24469413051552702745991592048, −5.406893184432130850004204558102, −4.663634105779138446653422573644, −3.76859933821994745432969456832, −2.81746290479980055360478807366, −1.66596544537524791242159006331, −1.05602146440879549653772852772, 0.11966239546688482892486123596, 0.65244541954917368029445900805, 2.15381408511070729815639589644, 2.58482805845204578906811845939, 3.63491945104628180266816112133, 4.27464879577630447026666024231, 4.72613464550844535930594529841, 5.33796203588789854852406482114, 7.02813294749861207293230543575, 7.749015597658145799704997622122, 8.22534364529064787560909727307, 8.90052529681647540743676722164, 9.768784881407148499346336969370, 10.384937837292500360338721197598, 10.89470124557791934411176009816, 11.65184066559116152849670738606, 12.06068079718735556994270756328, 13.07151357871484787933875675844, 13.75370055399119699761061290721, 14.543116291575981392934865185139, 15.349040882942902078482319681589, 15.62901258918664274968083481283, 16.681800697027580854809646327633, 17.39329431603892244192005866436, 17.794617022988953429050559021537

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