↑

Properties

Label 1-1045-1045.167-r1-0-0
Degree $1$
Conductor $1045$
Sign $-0.283 - 0.959i$
Analytic cond. $112.300$
Root an. cond. $112.300$
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  + (0.970 − 0.241i)2-s + (−0.999 + 0.0348i)3-s + (0.882 − 0.469i)4-s + (−0.961 + 0.275i)6-s + (−0.743 − 0.669i)7-s + (0.743 − 0.669i)8-s + (0.997 − 0.0697i)9-s + (−0.866 + 0.5i)12-s + (0.829 − 0.559i)13-s + (−0.882 − 0.469i)14-s + (0.559 − 0.829i)16-s + (−0.0697 + 0.997i)17-s + (0.951 − 0.309i)18-s + (0.766 + 0.642i)21-s + (0.984 − 0.173i)23-s + (−0.719 + 0.694i)24-s + ⋯
L(s)  = 1  + (0.970 − 0.241i)2-s + (−0.999 + 0.0348i)3-s + (0.882 − 0.469i)4-s + (−0.961 + 0.275i)6-s + (−0.743 − 0.669i)7-s + (0.743 − 0.669i)8-s + (0.997 − 0.0697i)9-s + (−0.866 + 0.5i)12-s + (0.829 − 0.559i)13-s + (−0.882 − 0.469i)14-s + (0.559 − 0.829i)16-s + (−0.0697 + 0.997i)17-s + (0.951 − 0.309i)18-s + (0.766 + 0.642i)21-s + (0.984 − 0.173i)23-s + (−0.719 + 0.694i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.283 - 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.283 - 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $-0.283 - 0.959i$
Analytic conductor: \(112.300\)
Root analytic conductor: \(112.300\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1045} (167, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1045,\ (1:\ ),\ -0.283 - 0.959i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.662168445 - 2.223525630i\)
\(L(\frac12)\) \(\approx\) \(1.662168445 - 2.223525630i\)
\(L(1)\) \(\approx\) \(1.368878505 - 0.5319833403i\)
\(L(1)\) \(\approx\) \(1.368878505 - 0.5319833403i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
19 \( 1 \)
good2 \( 1 + (0.970 - 0.241i)T \)
3 \( 1 + (-0.999 + 0.0348i)T \)
7 \( 1 + (-0.743 - 0.669i)T \)
13 \( 1 + (0.829 - 0.559i)T \)
17 \( 1 + (-0.0697 + 0.997i)T \)
23 \( 1 + (0.984 - 0.173i)T \)
29 \( 1 + (-0.848 - 0.529i)T \)
31 \( 1 + (0.104 - 0.994i)T \)
37 \( 1 + (0.951 - 0.309i)T \)
41 \( 1 + (0.0348 + 0.999i)T \)
43 \( 1 + (0.984 + 0.173i)T \)
47 \( 1 + (0.927 - 0.374i)T \)
53 \( 1 + (0.898 + 0.438i)T \)
59 \( 1 + (-0.374 + 0.927i)T \)
61 \( 1 + (0.719 + 0.694i)T \)
67 \( 1 + (-0.642 - 0.766i)T \)
71 \( 1 + (-0.438 - 0.898i)T \)
73 \( 1 + (0.788 - 0.615i)T \)
79 \( 1 + (-0.961 - 0.275i)T \)
83 \( 1 + (-0.406 + 0.913i)T \)
89 \( 1 + (-0.939 - 0.342i)T \)
97 \( 1 + (-0.970 + 0.241i)T \)
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

−21.7948255134546643700253815336, −21.10292854845712737152910619412, −20.29959255497599179260233885908, −19.09430781423983575844479922014, −18.51332779912761190353613146479, −17.488779022414920950946288603687, −16.63451845037895172449711088126, −15.95836382251132278731007787474, −15.58261286761519665487702694280, −14.47965558384968963865563438079, −13.49597848621569016292983257624, −12.84413995205775206092058771184, −12.152062831704955910748920549738, −11.37999910363157838608087719705, −10.79697369240809181161844170488, −9.60510751986620911785909992806, −8.67236391337062193592705494840, −7.22286417464356958587169022759, −6.79210716074603337523459462103, −5.830579908193303687570991015845, −5.31320074196818465169058682082, −4.31331316464769754852303791570, −3.37715337557456809707332216513, −2.32079904256686134888329474575, −1.05465758571815107703626543394, 0.53353899421275023367525806002, 1.38928669745060982043160279179, 2.7874514470928500799608129913, 3.910985618085187711449357740907, 4.35418785599765467453152927173, 5.711544565563790088481063677609, 6.03507014023394239085402340948, 6.963254688181062505039396672952, 7.77215148162821640521025958901, 9.364704326805453310155659445005, 10.31984221215856053014220256838, 10.845835546104724971040200717355, 11.52472617582370483394143009812, 12.62609916194137077367824560321, 13.04610617698809622871923532639, 13.680546537060681882640350913033, 15.04526297048565726944550343858, 15.44809242605938909289713942090, 16.63401369503625605811400987107, 16.753489989772873249517118567652, 17.9945320846894122120132553631, 18.93307901849698949877930935370, 19.63423571412841803593811512296, 20.60065937079595852909353482234, 21.203886652677033358574561950643

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