↑

Properties

Label 1-1045-1045.1017-r0-0-0
Degree $1$
Conductor $1045$
Sign $-0.999 - 0.0422i$
Analytic cond. $4.85295$
Root an. cond. $4.85295$
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  + (−0.829 − 0.559i)2-s + (0.139 + 0.990i)3-s + (0.374 + 0.927i)4-s + (0.438 − 0.898i)6-s + (−0.207 − 0.978i)7-s + (0.207 − 0.978i)8-s + (−0.961 + 0.275i)9-s + (−0.866 + 0.5i)12-s + (0.694 − 0.719i)13-s + (−0.374 + 0.927i)14-s + (−0.719 + 0.694i)16-s + (−0.275 + 0.961i)17-s + (0.951 + 0.309i)18-s + (0.939 − 0.342i)21-s + (−0.642 − 0.766i)23-s + (0.997 + 0.0697i)24-s + ⋯
L(s)  = 1  + (−0.829 − 0.559i)2-s + (0.139 + 0.990i)3-s + (0.374 + 0.927i)4-s + (0.438 − 0.898i)6-s + (−0.207 − 0.978i)7-s + (0.207 − 0.978i)8-s + (−0.961 + 0.275i)9-s + (−0.866 + 0.5i)12-s + (0.694 − 0.719i)13-s + (−0.374 + 0.927i)14-s + (−0.719 + 0.694i)16-s + (−0.275 + 0.961i)17-s + (0.951 + 0.309i)18-s + (0.939 − 0.342i)21-s + (−0.642 − 0.766i)23-s + (0.997 + 0.0697i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0422i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0422i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $-0.999 - 0.0422i$
Analytic conductor: \(4.85295\)
Root analytic conductor: \(4.85295\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1045} (1017, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1045,\ (0:\ ),\ -0.999 - 0.0422i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(4.530265496\times10^{-5} + 0.002142160073i\)
\(L(\frac12)\) \(\approx\) \(4.530265496\times10^{-5} + 0.002142160073i\)
\(L(1)\) \(\approx\) \(0.5735449570 + 0.0008704409192i\)
\(L(1)\) \(\approx\) \(0.5735449570 + 0.0008704409192i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
19 \( 1 \)
good2 \( 1 + (-0.829 - 0.559i)T \)
3 \( 1 + (0.139 + 0.990i)T \)
7 \( 1 + (-0.207 - 0.978i)T \)
13 \( 1 + (0.694 - 0.719i)T \)
17 \( 1 + (-0.275 + 0.961i)T \)
23 \( 1 + (-0.642 - 0.766i)T \)
29 \( 1 + (-0.615 + 0.788i)T \)
31 \( 1 + (-0.913 - 0.406i)T \)
37 \( 1 + (-0.951 - 0.309i)T \)
41 \( 1 + (-0.990 + 0.139i)T \)
43 \( 1 + (0.642 - 0.766i)T \)
47 \( 1 + (0.999 + 0.0348i)T \)
53 \( 1 + (-0.970 - 0.241i)T \)
59 \( 1 + (0.0348 + 0.999i)T \)
61 \( 1 + (-0.997 + 0.0697i)T \)
67 \( 1 + (-0.342 + 0.939i)T \)
71 \( 1 + (0.241 + 0.970i)T \)
73 \( 1 + (-0.469 + 0.882i)T \)
79 \( 1 + (0.438 + 0.898i)T \)
83 \( 1 + (0.994 + 0.104i)T \)
89 \( 1 + (0.173 + 0.984i)T \)
97 \( 1 + (-0.829 - 0.559i)T \)
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

−22.069541803365212605284534118775, −20.85509083895869283949149376961, −20.14249105083778964543048700461, −19.19978940152199133121383378214, −18.75638523168707858009073839777, −18.12431563599898318234788902712, −17.48020615945532633951297569870, −16.4837464946393743337259374700, −15.75903924306528854559668696487, −15.00020216103524019879437740718, −14.005760963398199895524172890196, −13.48907351714377756746296144753, −12.224811881703431963928214802835, −11.629064271736905780942831920872, −10.825223648695143598938721876908, −9.3575197451456249493153022008, −9.11601825867732557784334365761, −8.13099251107289663429245324062, −7.40185559901034904660611072479, −6.49331682948906939062340688988, −5.92566264415420499524005198966, −5.01071425008491306853594460778, −3.31371421113206916625678066148, −2.16695005429011355435152424681, −1.53718829690726479789127740381, 0.00114011393028290930313107339, 1.44150703408179793963293561934, 2.68553868806292062330025603383, 3.77293603324911511135248464745, 4.02319332298482878143716996234, 5.47166719326733193931910617069, 6.58032962018341934229248913959, 7.64037071340279908133988369803, 8.45508802584495759984997667235, 9.12872991951807550479432807487, 10.152401600027535502957634915085, 10.61778252183085915502850501592, 11.106810197477681719611547742829, 12.31827178618489739965303927685, 13.15115193739595535716473175519, 14.03391414069492509108802629200, 15.09402615646805492140883741813, 15.89147437202669662436298921264, 16.59432598367974331721663014102, 17.18985467814441400293016725255, 17.96066287727137738166032068170, 18.960688455079256928393596425773, 19.82324194431950111669769649215, 20.44291589038200754548097369348, 20.75990513998080971810023644318

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