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Properties

Label 1-4033-4033.56-r0-0-0
Degree $1$
Conductor $4033$
Sign $-0.859 - 0.511i$
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 + (0.686 − 0.727i)3-s + (−0.5 − 0.866i)4-s + (0.549 − 0.835i)5-s + (0.286 + 0.957i)6-s + (−0.0581 + 0.998i)7-s + 8-s + (−0.0581 − 0.998i)9-s + (0.448 + 0.893i)10-s + (−0.448 + 0.893i)11-s + (−0.973 − 0.230i)12-s + (−0.835 − 0.549i)13-s + (−0.835 − 0.549i)14-s + (−0.230 − 0.973i)15-s + (−0.5 + 0.866i)16-s − 17-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)2-s + (0.686 − 0.727i)3-s + (−0.5 − 0.866i)4-s + (0.549 − 0.835i)5-s + (0.286 + 0.957i)6-s + (−0.0581 + 0.998i)7-s + 8-s + (−0.0581 − 0.998i)9-s + (0.448 + 0.893i)10-s + (−0.448 + 0.893i)11-s + (−0.973 − 0.230i)12-s + (−0.835 − 0.549i)13-s + (−0.835 − 0.549i)14-s + (−0.230 − 0.973i)15-s + (−0.5 + 0.866i)16-s − 17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.859 - 0.511i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.859 - 0.511i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $-0.859 - 0.511i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (56, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ -0.859 - 0.511i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1389087220 - 0.5047915029i\)
\(L(\frac12)\) \(\approx\) \(0.1389087220 - 0.5047915029i\)
\(L(1)\) \(\approx\) \(0.8463223867 - 0.03884677393i\)
\(L(1)\) \(\approx\) \(0.8463223867 - 0.03884677393i\)

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 \( 1 + (0.686 - 0.727i)T \)
5 \( 1 + (0.549 - 0.835i)T \)
7 \( 1 + (-0.0581 + 0.998i)T \)
11 \( 1 + (-0.448 + 0.893i)T \)
13 \( 1 + (-0.835 - 0.549i)T \)
17 \( 1 - T \)
19 \( 1 + (-0.939 - 0.342i)T \)
23 \( 1 + (0.939 + 0.342i)T \)
29 \( 1 + (0.549 - 0.835i)T \)
31 \( 1 + (0.802 + 0.597i)T \)
41 \( 1 + (0.642 - 0.766i)T \)
43 \( 1 + (0.642 - 0.766i)T \)
47 \( 1 + (-0.957 - 0.286i)T \)
53 \( 1 + (-0.727 - 0.686i)T \)
59 \( 1 + (0.973 - 0.230i)T \)
61 \( 1 + (-0.116 - 0.993i)T \)
67 \( 1 + (0.230 + 0.973i)T \)
71 \( 1 + (0.5 + 0.866i)T \)
73 \( 1 + (0.835 - 0.549i)T \)
79 \( 1 + (-0.286 + 0.957i)T \)
83 \( 1 + (-0.396 + 0.918i)T \)
89 \( 1 + (-0.549 + 0.835i)T \)
97 \( 1 + (-0.802 + 0.597i)T \)
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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

−19.11987590320699777018889118854, −18.15549270167876154484881931587, −17.416430272530641110849368442063, −16.77661459760274258677073264910, −16.2914930163760799360664888968, −15.27586847700214152396987412525, −14.39446439941981514493443491487, −14.04440267529309141631277408777, −13.25085183922543693235748580285, −12.84358975028813855177467447247, −11.43980083480981616307019397274, −10.921017140260639772123087058205, −10.52358137984815401022389836157, −9.85401743839531626302289399734, −9.254576482482740740226744692161, −8.5002868865226828594924428317, −7.776058902081012016720280088742, −7.03546686395749347697815120109, −6.22269548249090849667827027155, −4.817351752075643732997425340305, −4.40571157791895649491439341834, −3.491354636666847402928114340027, −2.80237753317730376642862856194, −2.33473784098216957492757408370, −1.30212577768910736663288064236, 0.15274938897787176699257911475, 1.25752754010757244614456971854, 2.34811716228137116911858396878, 2.412606517740885247733793807350, 4.13190551403340465315278971578, 5.00252427194728673477603046037, 5.44498444462913776636351228653, 6.5237140903625715974005946778, 6.84430234296515484035441578705, 7.92812405047794178521427931051, 8.36739968659338880113919795852, 9.02232902275876798989391608742, 9.56654915597449608192646209618, 10.16990859895678786260507274144, 11.332197525836388952569462989298, 12.53438456886467895391080581583, 12.68297966472165714162547332225, 13.433044058807937567436528201102, 14.17013788876033138581267605166, 15.01821853727051570571893526221, 15.38464187992846753009423901182, 15.954343059353630870969697087530, 17.20979043995503093710116826129, 17.56452173693686573494914289992, 17.87857514058375677438910992861

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