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Properties

Label 1-571-571.169-r0-0-0
Degree $1$
Conductor $571$
Sign $-0.927 - 0.374i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.716 + 0.697i)2-s + (0.0935 + 0.995i)3-s + (0.0275 + 0.999i)4-s + (0.266 + 0.963i)5-s + (−0.627 + 0.778i)6-s + (−0.340 + 0.940i)7-s + (−0.677 + 0.735i)8-s + (−0.982 + 0.186i)9-s + (−0.480 + 0.876i)10-s + (0.999 + 0.0220i)11-s + (−0.992 + 0.120i)12-s + (0.00551 − 0.999i)13-s + (−0.899 + 0.436i)14-s + (−0.934 + 0.355i)15-s + (−0.998 + 0.0550i)16-s + (0.840 + 0.542i)17-s + ⋯
L(s)  = 1  + (0.716 + 0.697i)2-s + (0.0935 + 0.995i)3-s + (0.0275 + 0.999i)4-s + (0.266 + 0.963i)5-s + (−0.627 + 0.778i)6-s + (−0.340 + 0.940i)7-s + (−0.677 + 0.735i)8-s + (−0.982 + 0.186i)9-s + (−0.480 + 0.876i)10-s + (0.999 + 0.0220i)11-s + (−0.992 + 0.120i)12-s + (0.00551 − 0.999i)13-s + (−0.899 + 0.436i)14-s + (−0.934 + 0.355i)15-s + (−0.998 + 0.0550i)16-s + (0.840 + 0.542i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.927 - 0.374i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.927 - 0.374i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(571\)
Sign: $-0.927 - 0.374i$
Analytic conductor: \(2.65171\)
Root analytic conductor: \(2.65171\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{571} (169, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 571,\ (0:\ ),\ -0.927 - 0.374i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.3899121914 + 2.006478614i\)
\(L(\frac12)\) \(\approx\) \(-0.3899121914 + 2.006478614i\)
\(L(1)\) \(\approx\) \(0.7325193790 + 1.382406294i\)
\(L(1)\) \(\approx\) \(0.7325193790 + 1.382406294i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad571 \( 1 \)
good2 \( 1 + (0.716 + 0.697i)T \)
3 \( 1 + (0.0935 + 0.995i)T \)
5 \( 1 + (0.266 + 0.963i)T \)
7 \( 1 + (-0.340 + 0.940i)T \)
11 \( 1 + (0.999 + 0.0220i)T \)
13 \( 1 + (0.00551 - 0.999i)T \)
17 \( 1 + (0.840 + 0.542i)T \)
19 \( 1 + (-0.917 + 0.396i)T \)
23 \( 1 + (0.490 - 0.871i)T \)
29 \( 1 + (0.993 + 0.110i)T \)
31 \( 1 + (0.546 + 0.837i)T \)
37 \( 1 + (0.952 + 0.303i)T \)
41 \( 1 + (0.350 - 0.936i)T \)
43 \( 1 + (-0.126 - 0.991i)T \)
47 \( 1 + (0.451 - 0.892i)T \)
53 \( 1 + (-0.989 + 0.142i)T \)
59 \( 1 + (0.789 + 0.614i)T \)
61 \( 1 + (-0.556 - 0.831i)T \)
67 \( 1 + (0.618 + 0.785i)T \)
71 \( 1 + (0.669 - 0.743i)T \)
73 \( 1 + (-0.739 + 0.673i)T \)
79 \( 1 + (-0.234 + 0.972i)T \)
83 \( 1 + (0.988 + 0.153i)T \)
89 \( 1 + (-0.360 - 0.932i)T \)
97 \( 1 + (-0.609 - 0.792i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−23.141912481560794905466687589106, −21.86881558835466759517820706048, −21.01882119639493307661495356083, −20.26186633392915398003072672954, −19.40203952403297838940809644554, −19.178479000826068870020239134914, −17.7647706997421493937361838724, −16.973517498886624672701015083130, −16.2046513468571271086807457276, −14.69043442880785881534713034245, −13.96720107195471236659238354326, −13.36643774173974597409838389750, −12.65322162519303439126185219499, −11.815198820473820708472174070212, −11.159501455677080125252869342833, −9.68054057947018509366392710150, −9.18578689881468674734212335798, −7.85845035941902594645588664165, −6.6563786569159895879093636072, −6.10478559813451924187159424036, −4.770889491987825019701842231086, −3.985872116439820058057084746863, −2.749082364585297395055710196879, −1.49757530522075937946204472109, −0.88371253178235062007037293164, 2.458317722896763298148026169862, 3.203776899976851522226923235162, 4.02825494827388984743816661107, 5.2612001003836024874221596413, 6.03396557110987659206193184175, 6.70600470691193045850995155600, 8.15595997643874605368619428589, 8.83571589267525655767073984081, 9.98063198437376646862135736645, 10.79175956524422609893964348889, 11.92421787527793511642828743880, 12.64148925404502051994010809158, 13.97195944829409813550978140906, 14.66095727542783831514036712940, 15.1252848513158498892521872629, 15.87143886710561936464366594357, 16.90602256968212034425408556509, 17.514312095753935903937803206197, 18.65238308190534860117320741379, 19.62645072923280875281704155231, 20.79440177842644926981848644623, 21.602502858863153251428588788581, 22.05694650941420315317849612157, 22.77169596533715716005958442621, 23.32971314954058870982462761374

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