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Properties

Label 1-571-571.65-r0-0-0
Degree $1$
Conductor $571$
Sign $0.652 + 0.757i$
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  + (0.945 + 0.324i)2-s + (−0.627 + 0.778i)3-s + (0.789 + 0.614i)4-s + (0.980 + 0.197i)5-s + (−0.846 + 0.533i)6-s + (−0.461 − 0.887i)7-s + (0.546 + 0.837i)8-s + (−0.213 − 0.976i)9-s + (0.863 + 0.504i)10-s + (0.863 − 0.504i)11-s + (−0.973 + 0.229i)12-s + (0.991 − 0.131i)13-s + (−0.148 − 0.988i)14-s + (−0.768 + 0.639i)15-s + (0.245 + 0.969i)16-s + (0.371 − 0.928i)17-s + ⋯
L(s)  = 1  + (0.945 + 0.324i)2-s + (−0.627 + 0.778i)3-s + (0.789 + 0.614i)4-s + (0.980 + 0.197i)5-s + (−0.846 + 0.533i)6-s + (−0.461 − 0.887i)7-s + (0.546 + 0.837i)8-s + (−0.213 − 0.976i)9-s + (0.863 + 0.504i)10-s + (0.863 − 0.504i)11-s + (−0.973 + 0.229i)12-s + (0.991 − 0.131i)13-s + (−0.148 − 0.988i)14-s + (−0.768 + 0.639i)15-s + (0.245 + 0.969i)16-s + (0.371 − 0.928i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.652 + 0.757i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.652 + 0.757i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(571\)
Sign: $0.652 + 0.757i$
Analytic conductor: \(2.65171\)
Root analytic conductor: \(2.65171\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{571} (65, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 571,\ (0:\ ),\ 0.652 + 0.757i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.301594088 + 1.055217816i\)
\(L(\frac12)\) \(\approx\) \(2.301594088 + 1.055217816i\)
\(L(1)\) \(\approx\) \(1.730578081 + 0.6091150512i\)
\(L(1)\) \(\approx\) \(1.730578081 + 0.6091150512i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad571 \( 1 \)
good2 \( 1 + (0.945 + 0.324i)T \)
3 \( 1 + (-0.627 + 0.778i)T \)
5 \( 1 + (0.980 + 0.197i)T \)
7 \( 1 + (-0.461 - 0.887i)T \)
11 \( 1 + (0.863 - 0.504i)T \)
13 \( 1 + (0.991 - 0.131i)T \)
17 \( 1 + (0.371 - 0.928i)T \)
19 \( 1 + (-0.934 - 0.355i)T \)
23 \( 1 + (0.965 + 0.261i)T \)
29 \( 1 + (-0.879 - 0.475i)T \)
31 \( 1 + (0.245 + 0.969i)T \)
37 \( 1 + (0.431 - 0.901i)T \)
41 \( 1 + (-0.677 - 0.735i)T \)
43 \( 1 + (-0.995 + 0.0990i)T \)
47 \( 1 + (0.245 + 0.969i)T \)
53 \( 1 + (-0.956 - 0.293i)T \)
59 \( 1 + (-0.986 + 0.164i)T \)
61 \( 1 + (-0.0165 + 0.999i)T \)
67 \( 1 + (-0.956 - 0.293i)T \)
71 \( 1 + (0.309 + 0.951i)T \)
73 \( 1 + (0.431 - 0.901i)T \)
79 \( 1 + (0.828 + 0.560i)T \)
83 \( 1 + (-0.846 + 0.533i)T \)
89 \( 1 + (-0.846 + 0.533i)T \)
97 \( 1 + (-0.999 - 0.0330i)T \)
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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

−23.08986354710827098098317012674, −22.23790161652198562523644878321, −21.73539378966452062721190298824, −20.86626724226562094496590746142, −19.87155043393216086644740544797, −18.8082562026683628962768730352, −18.459563781324280185917690516516, −16.977984749451291742657560439814, −16.71441880609241866261115697998, −15.28001596340351601381679558824, −14.57126885399499981338453418453, −13.43447300496653600134852453697, −12.93142841887783063216497957231, −12.27790065192630280192293527700, −11.3798523060904962032034322375, −10.47847155480078885555243795699, −9.46285164040097536340573035061, −8.33173085105908656234449607855, −6.71122345288721973020152494211, −6.276937301969273805764832540384, −5.61975755971884728493206200604, −4.58388117152090542655410761460, −3.218091698977032607170192645072, −1.95370392988383993754961785489, −1.44791503792807942341358087240, 1.26744938642455508675350521479, 3.00434305423064308624832177497, 3.73869984308794253494978976020, 4.728449226187680781294701360637, 5.726797953576513602709246575552, 6.413454133563182533272335289337, 7.095260555286610983685650598855, 8.76139284631733037286921212848, 9.662198748769140476857932993861, 10.83016766883678825411073585738, 11.138334807464582678044843457879, 12.41558658975109420036549938322, 13.38590073676808924448766603600, 13.98891522003993494515725804316, 14.857558869440927119153092306440, 15.83111208943196029003878846803, 16.71923740690520528603367135255, 17.0291323874177553534198635767, 18.00491332577235524865914867916, 19.3939742863844515468917675794, 20.57582449126117986940026662854, 21.032148748418972467186233712076, 21.82515119020181132895335086304, 22.604925405130794388279505377561, 23.10176327519624175129212982777

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