Properties

Label 1-1840-1840.459-r0-0-0
Degree $1$
Conductor $1840$
Sign $0.923 - 0.382i$
Analytic cond. $8.54492$
Root an. cond. $8.54492$
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  i·3-s + 7-s − 9-s + i·11-s + i·13-s + 17-s i·19-s i·21-s + i·27-s i·29-s − 31-s + 33-s + i·37-s + 39-s − 41-s + ⋯
L(s)  = 1  i·3-s + 7-s − 9-s + i·11-s + i·13-s + 17-s i·19-s i·21-s + i·27-s i·29-s − 31-s + 33-s + i·37-s + 39-s − 41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.923 - 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.923 - 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1840\)    =    \(2^{4} \cdot 5 \cdot 23\)
Sign: $0.923 - 0.382i$
Analytic conductor: \(8.54492\)
Root analytic conductor: \(8.54492\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1840} (459, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1840,\ (0:\ ),\ 0.923 - 0.382i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.741004849 - 0.3463073961i\)
\(L(\frac12)\) \(\approx\) \(1.741004849 - 0.3463073961i\)
\(L(1)\) \(\approx\) \(1.169250902 - 0.2413364714i\)
\(L(1)\) \(\approx\) \(1.169250902 - 0.2413364714i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
23 \( 1 \)
good3 \( 1 + T \)
7 \( 1 - iT \)
11 \( 1 \)
13 \( 1 \)
17 \( 1 \)
19 \( 1 + T \)
29 \( 1 - T \)
31 \( 1 \)
37 \( 1 + iT \)
41 \( 1 \)
43 \( 1 + iT \)
47 \( 1 \)
53 \( 1 \)
59 \( 1 \)
61 \( 1 + T \)
67 \( 1 \)
71 \( 1 - iT \)
73 \( 1 \)
79 \( 1 - iT \)
83 \( 1 \)
89 \( 1 \)
97 \( 1 \)
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

−20.33045964435108627654918161541, −19.66867154146671119498375942677, −18.58016579153359533723444694822, −18.02338853956214495300547325923, −17.03857435337689659436401169575, −16.54572691435157544524228765954, −15.830641185077471937423211758881, −14.908841246158808849048665414787, −14.44546278175566270340118366732, −13.822362330390579418060371862385, −12.6512235665619038258985193833, −11.87120480243745701772379393969, −10.952414000591824395847789513380, −10.62287674002726735686580561454, −9.74295029809876526282500317680, −8.803516444146587304861685574410, −8.169801919321091148160564228212, −7.534347696914007719348341321982, −6.04889882309016312684168200765, −5.452178533829341308884003766382, −4.87547120854572617787042746598, −3.63135339906726101402173107616, −3.30827246018607069516930271592, −2.00695348881822200575862918754, −0.78792235225692061088831548334, 0.95374682248521726260129771078, 1.876701811228682016728757609666, 2.42920133697521020313651132969, 3.741999267228939077587473976744, 4.76077128282832059930427487452, 5.43757160392510095384426544976, 6.50922881175463624451821047068, 7.231166630703275135066635288406, 7.77866864393054927809422060475, 8.66330572749147369206377628662, 9.40451268397440967041683971899, 10.45360780500137908935307351682, 11.4182484206372897438685749488, 11.8958583296285071513118962733, 12.555904799260570804071886083879, 13.55584430278094418956050283071, 14.0594056396132342465334444187, 14.840633602668489783432220967522, 15.476867919032282881321794612237, 16.862111671523470871746736256, 17.136469189024065817640424574144, 17.99395834523548499783456725593, 18.601680188708184032756198951831, 19.19737590864294420293131579497, 20.18978406612622857783243378310

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