↑

Properties

Label 1-4033-4033.743-r0-0-0
Degree $1$
Conductor $4033$
Sign $0.940 - 0.339i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + (−0.0581 + 0.998i)3-s + 4-s + (−0.396 + 0.918i)5-s + (0.0581 − 0.998i)6-s + (−0.993 + 0.116i)7-s − 8-s + (−0.993 − 0.116i)9-s + (0.396 − 0.918i)10-s + (0.396 + 0.918i)11-s + (−0.0581 + 0.998i)12-s + (−0.396 + 0.918i)13-s + (0.993 − 0.116i)14-s + (−0.893 − 0.448i)15-s + 16-s + (0.5 + 0.866i)17-s + ⋯
L(s)  = 1  − 2-s + (−0.0581 + 0.998i)3-s + 4-s + (−0.396 + 0.918i)5-s + (0.0581 − 0.998i)6-s + (−0.993 + 0.116i)7-s − 8-s + (−0.993 − 0.116i)9-s + (0.396 − 0.918i)10-s + (0.396 + 0.918i)11-s + (−0.0581 + 0.998i)12-s + (−0.396 + 0.918i)13-s + (0.993 − 0.116i)14-s + (−0.893 − 0.448i)15-s + 16-s + (0.5 + 0.866i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.940 - 0.339i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.940 - 0.339i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.940 - 0.339i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (743, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ 0.940 - 0.339i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.08998329654 + 0.01575404711i\)
\(L(\frac12)\) \(\approx\) \(0.08998329654 + 0.01575404711i\)
\(L(1)\) \(\approx\) \(0.3843759719 + 0.2857066697i\)
\(L(1)\) \(\approx\) \(0.3843759719 + 0.2857066697i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
109 \( 1 \)
good2 \( 1 - T \)
3 \( 1 + (-0.0581 + 0.998i)T \)
5 \( 1 + (-0.396 + 0.918i)T \)
7 \( 1 + (-0.993 + 0.116i)T \)
11 \( 1 + (0.396 + 0.918i)T \)
13 \( 1 + (-0.396 + 0.918i)T \)
17 \( 1 + (0.5 + 0.866i)T \)
19 \( 1 + (-0.173 - 0.984i)T \)
23 \( 1 + (-0.766 + 0.642i)T \)
29 \( 1 + (0.993 - 0.116i)T \)
31 \( 1 + (0.686 + 0.727i)T \)
41 \( 1 + (0.766 + 0.642i)T \)
43 \( 1 + (-0.173 + 0.984i)T \)
47 \( 1 + (-0.0581 - 0.998i)T \)
53 \( 1 + (-0.835 - 0.549i)T \)
59 \( 1 + (-0.893 - 0.448i)T \)
61 \( 1 + (-0.973 - 0.230i)T \)
67 \( 1 + (-0.0581 - 0.998i)T \)
71 \( 1 + T \)
73 \( 1 + (-0.993 - 0.116i)T \)
79 \( 1 + (-0.893 - 0.448i)T \)
83 \( 1 + (-0.286 - 0.957i)T \)
89 \( 1 + (-0.597 - 0.802i)T \)
97 \( 1 + (0.686 - 0.727i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−18.707951453186819483326879442025, −17.83446873284710426959537207930, −17.09787960530504649732510123690, −16.63074606549627506655470117079, −16.07772820494273921975081964043, −15.43960737351108237706134577159, −14.29583162143456687165650206487, −13.696065797349486233144717746552, −12.69930570374806964195184641668, −12.27478773894684370781890649021, −11.86023965172853996120030579986, −10.89711561122139352216898477658, −10.11348883008069122858781538324, −9.36697564135307688282549585051, −8.662957271088758390292594510657, −8.01203430963770649760743034994, −7.58855320838236149604091635937, −6.663662659769039228833532049547, −5.96153859068597990585606308819, −5.49869991150387480416102623281, −4.07064825290733745096363135921, −3.068106555036662723579925453397, −2.56861461486167223224444475441, −1.292975625333793859738037674773, −0.74843390122855521138902675958, 0.05139069497151830316991100064, 1.661602479043089627538217134886, 2.64406854514551678142837666081, 3.19414626796118281848321817134, 4.00540456094393688476927224944, 4.823348359447912397450366282446, 6.24181281026381174517276024192, 6.37727693057832619921307492310, 7.25664911019087411275976486613, 8.04584720690936709507035983988, 8.94101697759215440396326674971, 9.592383151696515863450829312, 10.04408054311212945894084067288, 10.56683415322007711174895633965, 11.477106747628113838259578552150, 11.90120768479332327781564357065, 12.66402423805640126920570991162, 14.015084309927651180923157096927, 14.60066245253045902556227464871, 15.34622524100478965742214881945, 15.70339961682189836266951268237, 16.36530676887637768294615688463, 17.1321743138523828665281460792, 17.62647799463635203922475250283, 18.414544621984330594243650558770

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