↑

Properties

Label 1-571-571.155-r0-0-0
Degree $1$
Conductor $571$
Sign $0.387 - 0.922i$
Analytic cond. $2.65171$
Root an. cond. $2.65171$
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  + (−0.962 − 0.272i)2-s + (0.802 − 0.596i)3-s + (0.851 + 0.523i)4-s + (0.930 + 0.366i)5-s + (−0.934 + 0.355i)6-s + (−0.999 + 0.0330i)7-s + (−0.677 − 0.735i)8-s + (0.287 − 0.957i)9-s + (−0.795 − 0.605i)10-s + (−0.126 − 0.991i)11-s + (0.996 − 0.0880i)12-s + (0.411 + 0.911i)13-s + (0.970 + 0.240i)14-s + (0.965 − 0.261i)15-s + (0.451 + 0.892i)16-s + (0.988 − 0.153i)17-s + ⋯
L(s)  = 1  + (−0.962 − 0.272i)2-s + (0.802 − 0.596i)3-s + (0.851 + 0.523i)4-s + (0.930 + 0.366i)5-s + (−0.934 + 0.355i)6-s + (−0.999 + 0.0330i)7-s + (−0.677 − 0.735i)8-s + (0.287 − 0.957i)9-s + (−0.795 − 0.605i)10-s + (−0.126 − 0.991i)11-s + (0.996 − 0.0880i)12-s + (0.411 + 0.911i)13-s + (0.970 + 0.240i)14-s + (0.965 − 0.261i)15-s + (0.451 + 0.892i)16-s + (0.988 − 0.153i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.387 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.387 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(571\)
Sign: $0.387 - 0.922i$
Analytic conductor: \(2.65171\)
Root analytic conductor: \(2.65171\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{571} (155, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 571,\ (0:\ ),\ 0.387 - 0.922i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.078029667 - 0.7166210459i\)
\(L(\frac12)\) \(\approx\) \(1.078029667 - 0.7166210459i\)
\(L(1)\) \(\approx\) \(0.9583067458 - 0.3319563690i\)
\(L(1)\) \(\approx\) \(0.9583067458 - 0.3319563690i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad571 \( 1 \)
good2 \( 1 + (-0.962 - 0.272i)T \)
3 \( 1 + (0.802 - 0.596i)T \)
5 \( 1 + (0.930 + 0.366i)T \)
7 \( 1 + (-0.999 + 0.0330i)T \)
11 \( 1 + (-0.126 - 0.991i)T \)
13 \( 1 + (0.411 + 0.911i)T \)
17 \( 1 + (0.988 - 0.153i)T \)
19 \( 1 + (-0.999 + 0.0110i)T \)
23 \( 1 + (0.980 - 0.197i)T \)
29 \( 1 + (-0.592 - 0.805i)T \)
31 \( 1 + (0.546 - 0.837i)T \)
37 \( 1 + (0.202 + 0.979i)T \)
41 \( 1 + (0.635 - 0.771i)T \)
43 \( 1 + (0.329 - 0.944i)T \)
47 \( 1 + (-0.998 - 0.0550i)T \)
53 \( 1 + (-0.0385 + 0.999i)T \)
59 \( 1 + (0.789 - 0.614i)T \)
61 \( 1 + (-0.989 + 0.142i)T \)
67 \( 1 + (0.884 - 0.466i)T \)
71 \( 1 + (-0.104 + 0.994i)T \)
73 \( 1 + (-0.949 - 0.314i)T \)
79 \( 1 + (0.565 - 0.824i)T \)
83 \( 1 + (0.775 + 0.631i)T \)
89 \( 1 + (0.159 - 0.987i)T \)
97 \( 1 + (-0.234 + 0.972i)T \)
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

−23.42125434424170848600060777161, −22.602054828475745714761032118559, −21.24098558660787284505099480713, −20.9311857038634641859033574886, −19.901753470879238025500040354860, −19.41686317807220571058792482924, −18.35412041670310518977575618560, −17.51314082492824107123753235660, −16.60558610670654756173002401714, −16.02858674840681218401825160947, −15.022459296959672869295601759289, −14.459782173246570606029280885260, −13.09610323940632906611466457219, −12.61538766712078852882227452968, −10.81373665142095329828396717951, −10.160008598040047804944038188222, −9.56025515708139739566465681235, −8.87050308731491649136761544139, −7.92535444171149154504691687084, −6.90219281599464928039959175253, −5.858655878975472378930514346575, −4.9078944848370122337811527108, −3.32217784863409859699183131695, −2.458398910803300735166189067802, −1.31764650117760544201733818196, 0.91119632870074295336783070661, 2.113362641332472417433289583284, 2.89646572348939504655463317367, 3.72814326341338013862389050220, 6.02106137526168918185608423139, 6.47562209594134508825276608458, 7.42744497088538908661137534882, 8.54422140081366974648108467161, 9.22428617138889080874700355033, 9.895956033909725868259302500003, 10.86757480660284211748897469984, 11.959160300143018717344610200687, 13.02004403487367716662434610029, 13.553714513424478111466277775658, 14.60063779911463257054265150927, 15.60374145414937858335734812023, 16.70132456261692658513154295727, 17.213515978180521563673140195727, 18.59040114582402174025665257622, 18.862172411477903747357120594948, 19.312569039617505521969849376565, 20.67397003122991681938447935883, 21.10165371895783209686919255582, 21.93286896028146282126175693972, 23.19123504973890900157084976188

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