↑

Properties

Label 1-571-571.92-r0-0-0
Degree $1$
Conductor $571$
Sign $-0.829 + 0.558i$
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  + (−0.821 + 0.569i)2-s + (−0.556 + 0.831i)3-s + (0.350 − 0.936i)4-s + (0.775 + 0.631i)5-s + (−0.0165 − 0.999i)6-s + (−0.909 + 0.416i)7-s + (0.245 + 0.969i)8-s + (−0.381 − 0.924i)9-s + (−0.997 − 0.0770i)10-s + (0.565 + 0.824i)11-s + (0.583 + 0.812i)12-s + (0.970 + 0.240i)13-s + (0.509 − 0.860i)14-s + (−0.956 + 0.293i)15-s + (−0.754 − 0.656i)16-s + (0.996 + 0.0880i)17-s + ⋯
L(s)  = 1  + (−0.821 + 0.569i)2-s + (−0.556 + 0.831i)3-s + (0.350 − 0.936i)4-s + (0.775 + 0.631i)5-s + (−0.0165 − 0.999i)6-s + (−0.909 + 0.416i)7-s + (0.245 + 0.969i)8-s + (−0.381 − 0.924i)9-s + (−0.997 − 0.0770i)10-s + (0.565 + 0.824i)11-s + (0.583 + 0.812i)12-s + (0.970 + 0.240i)13-s + (0.509 − 0.860i)14-s + (−0.956 + 0.293i)15-s + (−0.754 − 0.656i)16-s + (0.996 + 0.0880i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.829 + 0.558i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.829 + 0.558i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(571\)
Sign: $-0.829 + 0.558i$
Analytic conductor: \(2.65171\)
Root analytic conductor: \(2.65171\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{571} (92, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 571,\ (0:\ ),\ -0.829 + 0.558i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2491465224 + 0.8153111560i\)
\(L(\frac12)\) \(\approx\) \(0.2491465224 + 0.8153111560i\)
\(L(1)\) \(\approx\) \(0.5401715881 + 0.4800802104i\)
\(L(1)\) \(\approx\) \(0.5401715881 + 0.4800802104i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad571 \( 1 \)
good2 \( 1 + (-0.821 + 0.569i)T \)
3 \( 1 + (-0.556 + 0.831i)T \)
5 \( 1 + (0.775 + 0.631i)T \)
7 \( 1 + (-0.909 + 0.416i)T \)
11 \( 1 + (0.565 + 0.824i)T \)
13 \( 1 + (0.970 + 0.240i)T \)
17 \( 1 + (0.996 + 0.0880i)T \)
19 \( 1 + (0.618 + 0.785i)T \)
23 \( 1 + (-0.846 - 0.533i)T \)
29 \( 1 + (0.137 - 0.990i)T \)
31 \( 1 + (0.945 - 0.324i)T \)
37 \( 1 + (0.528 + 0.849i)T \)
41 \( 1 + (-0.998 - 0.0550i)T \)
43 \( 1 + (0.761 + 0.648i)T \)
47 \( 1 + (-0.191 + 0.981i)T \)
53 \( 1 + (0.999 - 0.0220i)T \)
59 \( 1 + (-0.677 - 0.735i)T \)
61 \( 1 + (0.685 - 0.728i)T \)
67 \( 1 + (-0.480 + 0.876i)T \)
71 \( 1 + (0.669 + 0.743i)T \)
73 \( 1 + (0.471 - 0.882i)T \)
79 \( 1 + (-0.537 - 0.843i)T \)
83 \( 1 + (0.874 + 0.485i)T \)
89 \( 1 + (-0.857 + 0.514i)T \)
97 \( 1 + (-0.834 + 0.551i)T \)
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

−22.813644109820305326703160347691, −21.94805292589159612085819490697, −21.250914678994123649852349678366, −20.05861196456197428875459548967, −19.63914560730897256396237592565, −18.61131516881651934717037202212, −18.00412797887872739237311003322, −17.13741614289142981483719115428, −16.467057071572594201279844102614, −15.972240510802542996757737083523, −13.744500568583343718528947440866, −13.54836225265359806453629787948, −12.50277740979509095361702703622, −11.85390701052029743831739814944, −10.81482294228395993203816260791, −10.02176410216535930610419211843, −9.050587568884196073226504633273, −8.266216462348630736252928709597, −7.13956287983155072140083530627, −6.29305315589103282996158474646, −5.46513202206387012261113872193, −3.77642834189534898975406765228, −2.74685058188204373536888211883, −1.36788224299245902303521902775, −0.76203563087677703600605854353, 1.302582427793347780765521609295, 2.718946639689544052154007199, 3.96717963894414989576943282360, 5.37936299071880802907354006920, 6.24876508299145774097396675877, 6.51169623109502979174789797862, 7.98211029189881963929401136499, 9.20254652596064636038725649972, 9.91714371091682942527299211942, 10.153611898625058492427552347782, 11.39273614898131584041061059624, 12.22539168443004886988561708385, 13.74448212279106421674617930566, 14.585244919450162689808427068603, 15.358928446658236566325521894331, 16.16392716460612585070966250860, 16.836515994745800842936966400894, 17.64912083420815531910617258666, 18.43827680235674466607732574325, 19.08027600107053027986653828109, 20.36584681205867912273779038557, 20.99737652237534921427955730955, 22.14973785113788182733508286400, 22.78885989285548883473939507653, 23.33570279472500936743190986334

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