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Properties

Label 1-571-571.22-r1-0-0
Degree $1$
Conductor $571$
Sign $0.527 + 0.849i$
Analytic cond. $61.3624$
Root an. cond. $61.3624$
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  + (−0.546 + 0.837i)2-s + (−0.894 + 0.446i)3-s + (−0.401 − 0.915i)4-s + (0.828 − 0.560i)5-s + (0.115 − 0.993i)6-s + (−0.991 + 0.131i)7-s + (0.986 + 0.164i)8-s + (0.601 − 0.799i)9-s + (0.0165 + 0.999i)10-s + (−0.0165 + 0.999i)11-s + (0.768 + 0.639i)12-s + (0.922 + 0.386i)13-s + (0.431 − 0.901i)14-s + (−0.490 + 0.871i)15-s + (−0.677 + 0.735i)16-s + (0.909 + 0.416i)17-s + ⋯
L(s)  = 1  + (−0.546 + 0.837i)2-s + (−0.894 + 0.446i)3-s + (−0.401 − 0.915i)4-s + (0.828 − 0.560i)5-s + (0.115 − 0.993i)6-s + (−0.991 + 0.131i)7-s + (0.986 + 0.164i)8-s + (0.601 − 0.799i)9-s + (0.0165 + 0.999i)10-s + (−0.0165 + 0.999i)11-s + (0.768 + 0.639i)12-s + (0.922 + 0.386i)13-s + (0.431 − 0.901i)14-s + (−0.490 + 0.871i)15-s + (−0.677 + 0.735i)16-s + (0.909 + 0.416i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.527 + 0.849i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.527 + 0.849i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(571\)
Sign: $0.527 + 0.849i$
Analytic conductor: \(61.3624\)
Root analytic conductor: \(61.3624\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{571} (22, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 571,\ (1:\ ),\ 0.527 + 0.849i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9884360713 + 0.5494991119i\)
\(L(\frac12)\) \(\approx\) \(0.9884360713 + 0.5494991119i\)
\(L(1)\) \(\approx\) \(0.6528899177 + 0.2756880175i\)
\(L(1)\) \(\approx\) \(0.6528899177 + 0.2756880175i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad571 \( 1 \)
good2 \( 1 + (-0.546 + 0.837i)T \)
3 \( 1 + (-0.894 + 0.446i)T \)
5 \( 1 + (0.828 - 0.560i)T \)
7 \( 1 + (-0.991 + 0.131i)T \)
11 \( 1 + (-0.0165 + 0.999i)T \)
13 \( 1 + (0.922 + 0.386i)T \)
17 \( 1 + (0.909 + 0.416i)T \)
19 \( 1 + (0.461 - 0.887i)T \)
23 \( 1 + (0.701 - 0.712i)T \)
29 \( 1 + (-0.0825 + 0.996i)T \)
31 \( 1 + (-0.677 + 0.735i)T \)
37 \( 1 + (-0.973 - 0.229i)T \)
41 \( 1 + (-0.789 - 0.614i)T \)
43 \( 1 + (-0.956 - 0.293i)T \)
47 \( 1 + (0.677 - 0.735i)T \)
53 \( 1 + (0.627 - 0.778i)T \)
59 \( 1 + (-0.879 - 0.475i)T \)
61 \( 1 + (0.0495 + 0.998i)T \)
67 \( 1 + (0.627 - 0.778i)T \)
71 \( 1 + (0.809 - 0.587i)T \)
73 \( 1 + (0.973 + 0.229i)T \)
79 \( 1 + (0.213 + 0.976i)T \)
83 \( 1 + (0.115 - 0.993i)T \)
89 \( 1 + (-0.115 + 0.993i)T \)
97 \( 1 + (-0.995 + 0.0990i)T \)
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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

−22.82253444172793700971384277566, −22.02464773068753386551072729039, −21.34366226452578687818601179456, −20.477825534957506155908820778782, −19.16356116360019376404821562058, −18.693399513460429614185451789802, −18.2059445217469794151513685352, −17.0149317187198771092556102674, −16.68043804515976409146337884191, −15.664006772757580411869807027040, −13.82043511800957016704302810148, −13.46513065717996005479689459343, −12.59741311264982902215557200254, −11.596329346144625450119596160466, −10.86284742017896103019973132803, −10.093687495776009357874721223405, −9.421273157811688339722214924560, −8.09568358954321478859625689665, −7.12118594676473003325404359014, −6.09492873304591529243244292042, −5.42626135173700499275664887285, −3.65116833860426655164704863467, −2.95490782647506144534941396443, −1.566668376979323451643147557537, −0.676326525569177917680589612992, 0.65881161598219641860702969449, 1.72551313526011377002235190107, 3.662672005002563468378027906671, 5.00346966545934243173241156167, 5.457844704787774487197601566941, 6.59082197397872333305097234329, 6.97259114356995217141811378678, 8.72638415776414482025579114916, 9.29422171939112641984011345806, 10.12300803702583808263101853412, 10.721733910148146132955413290267, 12.21870274546537541301937803091, 12.950533197593337118395039267454, 13.9259687763602587006873191959, 15.0944538780027533709282471929, 15.83914779668666638877095088956, 16.63323352699414007550291896967, 17.030667964594296522491824782770, 18.075264824844984452083204466773, 18.52812311164431423049472409627, 19.80968295786728512876972519900, 20.70086199469645324312566048437, 21.71131045814575898977317619407, 22.52129607933890307952886074545, 23.273523824954374192120749489447

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