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Properties

Label 1-571-571.12-r1-0-0
Degree $1$
Conductor $571$
Sign $0.689 - 0.724i$
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  + (0.592 + 0.805i)2-s + (0.857 − 0.514i)3-s + (−0.298 + 0.954i)4-s + (−0.170 − 0.985i)5-s + (0.922 + 0.386i)6-s + (0.627 + 0.778i)7-s + (−0.945 + 0.324i)8-s + (0.471 − 0.882i)9-s + (0.693 − 0.720i)10-s + (0.970 − 0.240i)11-s + (0.234 + 0.972i)12-s + (−0.0605 − 0.998i)13-s + (−0.256 + 0.966i)14-s + (−0.652 − 0.757i)15-s + (−0.821 − 0.569i)16-s + (−0.999 + 0.0220i)17-s + ⋯
L(s)  = 1  + (0.592 + 0.805i)2-s + (0.857 − 0.514i)3-s + (−0.298 + 0.954i)4-s + (−0.170 − 0.985i)5-s + (0.922 + 0.386i)6-s + (0.627 + 0.778i)7-s + (−0.945 + 0.324i)8-s + (0.471 − 0.882i)9-s + (0.693 − 0.720i)10-s + (0.970 − 0.240i)11-s + (0.234 + 0.972i)12-s + (−0.0605 − 0.998i)13-s + (−0.256 + 0.966i)14-s + (−0.652 − 0.757i)15-s + (−0.821 − 0.569i)16-s + (−0.999 + 0.0220i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.689 - 0.724i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.689 - 0.724i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(571\)
Sign: $0.689 - 0.724i$
Analytic conductor: \(61.3624\)
Root analytic conductor: \(61.3624\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{571} (12, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 571,\ (1:\ ),\ 0.689 - 0.724i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.243385584 - 1.390679182i\)
\(L(\frac12)\) \(\approx\) \(3.243385584 - 1.390679182i\)
\(L(1)\) \(\approx\) \(1.869479499 + 0.03582066190i\)
\(L(1)\) \(\approx\) \(1.869479499 + 0.03582066190i\)

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 \( 1 + (0.857 - 0.514i)T \)
5 \( 1 + (-0.170 - 0.985i)T \)
7 \( 1 + (0.627 + 0.778i)T \)
11 \( 1 + (0.970 - 0.240i)T \)
13 \( 1 + (-0.0605 - 0.998i)T \)
17 \( 1 + (-0.999 + 0.0220i)T \)
19 \( 1 + (-0.224 - 0.974i)T \)
23 \( 1 + (0.601 - 0.799i)T \)
29 \( 1 + (0.350 - 0.936i)T \)
31 \( 1 + (-0.0825 + 0.996i)T \)
37 \( 1 + (-0.968 + 0.250i)T \)
41 \( 1 + (-0.716 - 0.697i)T \)
43 \( 1 + (0.984 - 0.175i)T \)
47 \( 1 + (-0.904 + 0.426i)T \)
53 \( 1 + (-0.00551 + 0.999i)T \)
59 \( 1 + (0.546 - 0.837i)T \)
61 \( 1 + (0.202 - 0.979i)T \)
67 \( 1 + (0.868 - 0.495i)T \)
71 \( 1 + (0.978 - 0.207i)T \)
73 \( 1 + (-0.266 + 0.963i)T \)
79 \( 1 + (-0.509 + 0.860i)T \)
83 \( 1 + (-0.126 - 0.991i)T \)
89 \( 1 + (0.795 - 0.605i)T \)
97 \( 1 + (0.802 - 0.596i)T \)
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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

−22.873242253549919400256712778426, −22.20228737918317319878498838923, −21.40444342450103796573105584329, −20.75978824988196628500147048461, −19.78293022717566361931128125029, −19.38296189277948146730625776018, −18.47790549974505615126869225979, −17.43310515282139808414152085196, −16.20901098632230260335676637713, −15.01369126040273574187415040044, −14.58179096218386487777856577559, −13.9648482676998199011455982128, −13.219632901969374633905700109864, −11.775351679309768014036681262489, −11.14835531983304244795649816083, −10.346210858999244213896565144428, −9.54793299879756615449192334119, −8.614109931533396972228675977052, −7.28620546625501098125343967218, −6.51045593294741738015607582588, −4.941954241141077947366982827900, −3.99012927712708054109635273992, −3.600133607249668661637042583386, −2.263304877608700341574890956919, −1.504839758464414514340468800657, 0.60299218188887431872353935068, 2.0586442937375859565034902655, 3.16182065382307111387149923233, 4.32609986863236022270496525484, 5.09433475675773032044032437682, 6.248242657488829105214034902815, 7.143147968066802396332898857984, 8.37693271682106125609866151300, 8.57256751469447185848676128805, 9.37564142765984394700264442913, 11.296214993165664280614977712068, 12.29658299542729687291605410781, 12.77056610987449531758974351985, 13.686861006021858791254733040081, 14.454397630889122365888083723125, 15.44580559281759173198569657915, 15.71701406049077492916710373091, 17.33086941363681218823208423825, 17.50703243837004681340224838571, 18.739195835608844535507001787995, 19.77099146312378012115472196960, 20.50380338235727238959433145052, 21.29463840361043204328597302816, 22.107253307286665627812328998491, 23.13283534746897656430937169021

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