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Properties

Label 1-571-571.377-r0-0-0
Degree $1$
Conductor $571$
Sign $-0.934 + 0.355i$
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  + (0.245 − 0.969i)2-s + (0.980 − 0.197i)3-s + (−0.879 − 0.475i)4-s + (−0.461 − 0.887i)5-s + (0.0495 − 0.998i)6-s + (−0.277 − 0.960i)7-s + (−0.677 + 0.735i)8-s + (0.922 − 0.386i)9-s + (−0.973 + 0.229i)10-s + (−0.973 − 0.229i)11-s + (−0.956 − 0.293i)12-s + (0.746 − 0.665i)13-s + (−0.999 + 0.0330i)14-s + (−0.627 − 0.778i)15-s + (0.546 + 0.837i)16-s + (0.965 − 0.261i)17-s + ⋯
L(s)  = 1  + (0.245 − 0.969i)2-s + (0.980 − 0.197i)3-s + (−0.879 − 0.475i)4-s + (−0.461 − 0.887i)5-s + (0.0495 − 0.998i)6-s + (−0.277 − 0.960i)7-s + (−0.677 + 0.735i)8-s + (0.922 − 0.386i)9-s + (−0.973 + 0.229i)10-s + (−0.973 − 0.229i)11-s + (−0.956 − 0.293i)12-s + (0.746 − 0.665i)13-s + (−0.999 + 0.0330i)14-s + (−0.627 − 0.778i)15-s + (0.546 + 0.837i)16-s + (0.965 − 0.261i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.934 + 0.355i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.934 + 0.355i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(571\)
Sign: $-0.934 + 0.355i$
Analytic conductor: \(2.65171\)
Root analytic conductor: \(2.65171\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{571} (377, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 571,\ (0:\ ),\ -0.934 + 0.355i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.2694187586 - 1.467518634i\)
\(L(\frac12)\) \(\approx\) \(-0.2694187586 - 1.467518634i\)
\(L(1)\) \(\approx\) \(0.7006793507 - 1.041474711i\)
\(L(1)\) \(\approx\) \(0.7006793507 - 1.041474711i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad571 \( 1 \)
good2 \( 1 + (0.245 - 0.969i)T \)
3 \( 1 + (0.980 - 0.197i)T \)
5 \( 1 + (-0.461 - 0.887i)T \)
7 \( 1 + (-0.277 - 0.960i)T \)
11 \( 1 + (-0.973 - 0.229i)T \)
13 \( 1 + (0.746 - 0.665i)T \)
17 \( 1 + (0.965 - 0.261i)T \)
19 \( 1 + (-0.909 - 0.416i)T \)
23 \( 1 + (0.115 + 0.993i)T \)
29 \( 1 + (-0.401 - 0.915i)T \)
31 \( 1 + (0.546 + 0.837i)T \)
37 \( 1 + (-0.995 - 0.0990i)T \)
41 \( 1 + (-0.986 + 0.164i)T \)
43 \( 1 + (-0.518 - 0.854i)T \)
47 \( 1 + (0.546 + 0.837i)T \)
53 \( 1 + (0.997 + 0.0660i)T \)
59 \( 1 + (0.789 + 0.614i)T \)
61 \( 1 + (-0.768 + 0.639i)T \)
67 \( 1 + (0.997 + 0.0660i)T \)
71 \( 1 + (-0.809 - 0.587i)T \)
73 \( 1 + (-0.995 - 0.0990i)T \)
79 \( 1 + (0.991 + 0.131i)T \)
83 \( 1 + (0.0495 - 0.998i)T \)
89 \( 1 + (0.0495 - 0.998i)T \)
97 \( 1 + (0.180 - 0.983i)T \)
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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

−23.679533823220272620481649547730, −23.06315285884573565116891942371, −22.054178220845155269329703689913, −21.38067275091058240053614365414, −20.62601899237856519683434751827, −19.04039165674070836231676164843, −18.7761444074391437235957725431, −18.168529640469614838419773248864, −16.6178733679901453597473832625, −15.90805100608175819267015000763, −15.10115256578213999487581790303, −14.75120686023699769916772413216, −13.80515612895364062628361569648, −12.895311379051864183304525308631, −12.07038068835901983186144238371, −10.5780989401264841442738236733, −9.75090025064771774533436599984, −8.5513056027043671837628309305, −8.198118391671337179409011902412, −7.097668066363630608709265552441, −6.30570855204266707525408670665, −5.1502965241277422691512348236, −3.95942424319160888900757418216, −3.207939750890239418607615026291, −2.19454681317100008646687758351, 0.64024138012906186199511531971, 1.64036290691455614664913098495, 3.05261571818341061537248885963, 3.67524365374353863301690597078, 4.60824458777065896479206334301, 5.67814878800731682183334718510, 7.333117018797305639440610235531, 8.20346755832031716648606912005, 8.85940482411175614210855966709, 9.98779364007218561766247459708, 10.57686668442089176915200423833, 11.81425432882104327095013080734, 12.77327591760869656890924627767, 13.38286643023981800400784096107, 13.81345379271794498060271122820, 15.14571904752974829484164007720, 15.8078279456890668824691820279, 17.02608773387044281529416489216, 18.02818097931157558318928709827, 19.1265883873620529801631737295, 19.46533636929116917314740971387, 20.56237508905324626073773235481, 20.71427310497892057255212269907, 21.52857022340985357269397611794, 23.058702791266384032803345660552

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