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Properties

Label 1-571-571.123-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: $\chi_{571} (123, \cdot )$
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

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

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