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Properties

Label 1-571-571.224-r0-0-0
Degree $1$
Conductor $571$
Sign $0.588 + 0.808i$
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  + (−0.592 + 0.805i)2-s + (0.391 + 0.920i)3-s + (−0.298 − 0.954i)4-s + (−0.989 + 0.142i)5-s + (−0.973 − 0.229i)6-s + (0.965 − 0.261i)7-s + (0.945 + 0.324i)8-s + (−0.693 + 0.720i)9-s + (0.471 − 0.882i)10-s + (0.528 − 0.849i)11-s + (0.761 − 0.648i)12-s + (−0.968 + 0.250i)13-s + (−0.360 + 0.932i)14-s + (−0.518 − 0.854i)15-s + (−0.821 + 0.569i)16-s + (0.329 − 0.944i)17-s + ⋯
L(s)  = 1  + (−0.592 + 0.805i)2-s + (0.391 + 0.920i)3-s + (−0.298 − 0.954i)4-s + (−0.989 + 0.142i)5-s + (−0.973 − 0.229i)6-s + (0.965 − 0.261i)7-s + (0.945 + 0.324i)8-s + (−0.693 + 0.720i)9-s + (0.471 − 0.882i)10-s + (0.528 − 0.849i)11-s + (0.761 − 0.648i)12-s + (−0.968 + 0.250i)13-s + (−0.360 + 0.932i)14-s + (−0.518 − 0.854i)15-s + (−0.821 + 0.569i)16-s + (0.329 − 0.944i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.588 + 0.808i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.588 + 0.808i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(571\)
Sign: $0.588 + 0.808i$
Analytic conductor: \(2.65171\)
Root analytic conductor: \(2.65171\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{571} (224, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 571,\ (0:\ ),\ 0.588 + 0.808i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8920529034 + 0.4541136302i\)
\(L(\frac12)\) \(\approx\) \(0.8920529034 + 0.4541136302i\)
\(L(1)\) \(\approx\) \(0.7511116517 + 0.3812065685i\)
\(L(1)\) \(\approx\) \(0.7511116517 + 0.3812065685i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad571 \( 1 \)
good2 \( 1 + (-0.592 + 0.805i)T \)
3 \( 1 + (0.391 + 0.920i)T \)
5 \( 1 + (-0.989 + 0.142i)T \)
7 \( 1 + (0.965 - 0.261i)T \)
11 \( 1 + (0.528 - 0.849i)T \)
13 \( 1 + (-0.968 + 0.250i)T \)
17 \( 1 + (0.329 - 0.944i)T \)
19 \( 1 + (0.996 - 0.0880i)T \)
23 \( 1 + (-0.0165 - 0.999i)T \)
29 \( 1 + (0.350 + 0.936i)T \)
31 \( 1 + (-0.0825 - 0.996i)T \)
37 \( 1 + (-0.0605 - 0.998i)T \)
41 \( 1 + (0.716 - 0.697i)T \)
43 \( 1 + (-0.899 + 0.436i)T \)
47 \( 1 + (0.904 + 0.426i)T \)
53 \( 1 + (0.952 + 0.303i)T \)
59 \( 1 + (0.546 + 0.837i)T \)
61 \( 1 + (0.411 - 0.911i)T \)
67 \( 1 + (-0.739 + 0.673i)T \)
71 \( 1 + (0.669 + 0.743i)T \)
73 \( 1 + (-0.834 + 0.551i)T \)
79 \( 1 + (0.0935 - 0.995i)T \)
83 \( 1 + (0.685 - 0.728i)T \)
89 \( 1 + (0.287 + 0.957i)T \)
97 \( 1 + (-0.319 + 0.947i)T \)
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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

−23.16199257784447254709529007879, −22.24665428404793988900179489801, −21.191414492983541277006343898944, −20.27072288838437112183549161035, −19.74134932737416862010960658789, −19.16744342289964975090783708262, −18.17896885571124860524771490910, −17.5559746076015744914185878078, −16.83667651008941975542933994492, −15.359454248176110941710634748445, −14.66221831457284088450799432505, −13.61126552800290939989813144689, −12.48438446815543828214840243095, −11.961635607602777542690008798897, −11.50593351594006480335717603250, −10.18010785889872697641323843526, −9.15011896643077570523057374005, −8.20153177740656228809987349647, −7.696384343728773825216248822119, −6.962865637229993891918774541150, −5.178995072678418277814940657270, −4.07184277541966469933876261842, −3.06208062520642358855091551382, −1.921268386218341127187284303189, −1.081598076363002140686274964677, 0.774170943820691550635990669364, 2.57651529443364114440047902841, 3.93519315757445069296289545051, 4.74134634976552489541978598075, 5.53621901023484484379593690837, 7.102912843373349504765439118975, 7.72178281974808086572131418146, 8.607213823352138044711411918282, 9.29326674918605733063177281391, 10.38847443378034514323454244576, 11.16949934772178693543404884726, 11.89979850324792297119688127726, 13.78136981672414090067927367172, 14.48636027235583291663597402460, 14.84478534621805944908395515483, 16.05509525956057790481578016951, 16.37350404629477043227256963146, 17.28469158105642252698041404083, 18.380992924844130141382861086782, 19.19765916009957030526040739590, 20.02152363493540633583123999782, 20.590256179331525733820560838918, 21.89863584166499313916873551943, 22.57782468191732459001442197266, 23.49925444218086384210771136951

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