Properties

Label 1-4033-4033.8-r0-0-0
Degree $1$
Conductor $4033$
Sign $0.960 + 0.277i$
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  + (−0.5 + 0.866i)2-s − 3-s + (−0.5 − 0.866i)4-s + i·5-s + (0.5 − 0.866i)6-s + 7-s + 8-s + 9-s + (−0.866 − 0.5i)10-s + (0.866 − 0.5i)11-s + (0.5 + 0.866i)12-s + 13-s + (−0.5 + 0.866i)14-s i·15-s + (−0.5 + 0.866i)16-s + (0.5 − 0.866i)17-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)2-s − 3-s + (−0.5 − 0.866i)4-s + i·5-s + (0.5 − 0.866i)6-s + 7-s + 8-s + 9-s + (−0.866 − 0.5i)10-s + (0.866 − 0.5i)11-s + (0.5 + 0.866i)12-s + 13-s + (−0.5 + 0.866i)14-s i·15-s + (−0.5 + 0.866i)16-s + (0.5 − 0.866i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.960 + 0.277i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.960 + 0.277i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.960 + 0.277i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (8, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ 0.960 + 0.277i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.070398434 + 0.1515882530i\)
\(L(\frac12)\) \(\approx\) \(1.070398434 + 0.1515882530i\)
\(L(1)\) \(\approx\) \(0.7041175762 + 0.2505407426i\)
\(L(1)\) \(\approx\) \(0.7041175762 + 0.2505407426i\)

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 \( 1 - T \)
5 \( 1 + iT \)
7 \( 1 + T \)
11 \( 1 + (0.866 - 0.5i)T \)
13 \( 1 + T \)
17 \( 1 + (0.5 - 0.866i)T \)
19 \( 1 + (-0.5 - 0.866i)T \)
23 \( 1 - T \)
29 \( 1 + (-0.866 - 0.5i)T \)
31 \( 1 + (0.866 - 0.5i)T \)
41 \( 1 + (-0.866 + 0.5i)T \)
43 \( 1 - iT \)
47 \( 1 + (0.866 - 0.5i)T \)
53 \( 1 + (0.866 - 0.5i)T \)
59 \( 1 + T \)
61 \( 1 + iT \)
67 \( 1 + (-0.866 - 0.5i)T \)
71 \( 1 + (0.5 + 0.866i)T \)
73 \( 1 + (0.5 + 0.866i)T \)
79 \( 1 + (-0.5 + 0.866i)T \)
83 \( 1 + (0.5 + 0.866i)T \)
89 \( 1 + (-0.866 + 0.5i)T \)
97 \( 1 + (-0.866 - 0.5i)T \)
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

−18.39222804349652400874223316961, −17.60503483886986404239908194955, −17.30246555479397741246349072057, −16.62546333065305316141037833407, −16.133614618078089127579161018759, −15.12312766087424090258168010867, −14.18371660348009408516663396428, −13.3787180186831355754692660901, −12.6412766969787245785059879464, −12.00902129604681252266667714951, −11.78410158311884526832886881577, −10.81592909232963873978427671019, −10.35674112334582489564101707739, −9.54118953521753207025028258297, −8.738723002810287560234791926420, −8.15121891819586784683564079247, −7.50203165702041448774633334653, −6.34747734963323477800995154261, −5.626322715211684574181413968585, −4.77807698320792876369470572618, −4.09810310008312271239036660062, −3.70512896176744069707882151552, −1.89313694973011737163959707097, −1.56050595311016038867634871253, −0.92503806600320912695758873637, 0.54836221163082018621219735643, 1.39934513336430726005802046851, 2.3268704433920142290118283441, 3.8693963149030983606509211155, 4.28990157080627850556758769388, 5.499250992489327752898025820386, 5.75660840851458414272389174899, 6.74289683429650731607977226455, 7.00674290008689806029656079283, 7.96516388236845665236702293039, 8.59562801740563836688451493316, 9.578630403350883338908667706773, 10.23001121141519537287104613466, 10.97550385029193018286552985572, 11.434412672361002848295023420381, 11.95500877730254905699404564117, 13.47388588154307577932610078523, 13.748819135721952975358973847589, 14.5771048433895237744545579632, 15.30430458800263501688559150362, 15.723218204875301474306518823018, 16.71518518518557509233444479495, 17.05741532052384517202834325069, 17.91440339051483797865438552338, 18.24143119352890674552578309595

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