↑

Properties

Label 1-4033-4033.1003-r0-0-0
Degree $1$
Conductor $4033$
Sign $-0.702 + 0.711i$
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  + (0.939 − 0.342i)2-s + (0.597 − 0.802i)3-s + (0.766 − 0.642i)4-s + (0.286 − 0.957i)5-s + (0.286 − 0.957i)6-s + (−0.686 − 0.727i)7-s + (0.5 − 0.866i)8-s + (−0.286 − 0.957i)9-s + (−0.0581 − 0.998i)10-s + (−0.0581 + 0.998i)11-s + (−0.0581 − 0.998i)12-s + (−0.973 + 0.230i)13-s + (−0.893 − 0.448i)14-s + (−0.597 − 0.802i)15-s + (0.173 − 0.984i)16-s + (0.939 − 0.342i)17-s + ⋯
L(s)  = 1  + (0.939 − 0.342i)2-s + (0.597 − 0.802i)3-s + (0.766 − 0.642i)4-s + (0.286 − 0.957i)5-s + (0.286 − 0.957i)6-s + (−0.686 − 0.727i)7-s + (0.5 − 0.866i)8-s + (−0.286 − 0.957i)9-s + (−0.0581 − 0.998i)10-s + (−0.0581 + 0.998i)11-s + (−0.0581 − 0.998i)12-s + (−0.973 + 0.230i)13-s + (−0.893 − 0.448i)14-s + (−0.597 − 0.802i)15-s + (0.173 − 0.984i)16-s + (0.939 − 0.342i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.702 + 0.711i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.702 + 0.711i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $-0.702 + 0.711i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (1003, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ -0.702 + 0.711i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.9841535452 - 2.356161362i\)
\(L(\frac12)\) \(\approx\) \(-0.9841535452 - 2.356161362i\)
\(L(1)\) \(\approx\) \(1.196352147 - 1.424120990i\)
\(L(1)\) \(\approx\) \(1.196352147 - 1.424120990i\)

Euler product

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

−19.12031026606888771327600240847, −18.274237195268717249428071155806, −17.25632505107352947731534422840, −16.63075122504803519308914544234, −15.894923021438038089726666731738, −15.33731018430422510723813672753, −14.90812559515792822482747875155, −14.12304462156142775674697261948, −13.69246839294838835928271681320, −13.02235427873487559540834070813, −11.92581594887955433527379071373, −11.55551961694981947616978285059, −10.490031188415585026681861068518, −10.08165848331448728076281196909, −9.21702760364004993741333187630, −8.393051658604622467863790681472, −7.59831623699605616344974025439, −6.95471380871103972634643411019, −5.88793985386938673665029322565, −5.61113052151808766317366702495, −4.76682910992167701963839665729, −3.57439535003883622019932272200, −3.28343500395585035331416647304, −2.65513123021767059705568642548, −1.93294330493798603055853484190, 0.38513816017144390260780651455, 1.540735107749001468274266426299, 1.879858393005663199928371105552, 2.980115099343130273568785842453, 3.60154335177500250334966656752, 4.50825864570065109609717289305, 5.12407415038222655175117677300, 6.08123286139185355150927173466, 6.71211387966182212885129952723, 7.53753884564066095646320901116, 7.93921946868098565002325137084, 9.28480706669820494354345980439, 9.8084457329081527377042489050, 10.21382536117545809622494076219, 11.59842358149727531510014594577, 12.25378886397511592053966582408, 12.6063681535797417828862800480, 13.128229189310044256021750859389, 13.913391263135170765727799147504, 14.41568776165124118755986217942, 14.969261778515704212836521166, 16.094657141691882063966774880008, 16.5404782292428239608220962357, 17.25385985004903775911505701031, 18.23514670198936157858831556165

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