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Properties

Label 1-571-571.389-r1-0-0
Degree $1$
Conductor $571$
Sign $0.996 - 0.0874i$
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  + (0.986 + 0.164i)2-s + (−0.991 + 0.131i)3-s + (0.945 + 0.324i)4-s + (0.746 + 0.665i)5-s + (−0.999 − 0.0330i)6-s + (−0.652 − 0.757i)7-s + (0.879 + 0.475i)8-s + (0.965 − 0.261i)9-s + (0.627 + 0.778i)10-s + (−0.627 + 0.778i)11-s + (−0.980 − 0.197i)12-s + (−0.846 − 0.533i)13-s + (−0.518 − 0.854i)14-s + (−0.828 − 0.560i)15-s + (0.789 + 0.614i)16-s + (0.340 − 0.940i)17-s + ⋯
L(s)  = 1  + (0.986 + 0.164i)2-s + (−0.991 + 0.131i)3-s + (0.945 + 0.324i)4-s + (0.746 + 0.665i)5-s + (−0.999 − 0.0330i)6-s + (−0.652 − 0.757i)7-s + (0.879 + 0.475i)8-s + (0.965 − 0.261i)9-s + (0.627 + 0.778i)10-s + (−0.627 + 0.778i)11-s + (−0.980 − 0.197i)12-s + (−0.846 − 0.533i)13-s + (−0.518 − 0.854i)14-s + (−0.828 − 0.560i)15-s + (0.789 + 0.614i)16-s + (0.340 − 0.940i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.996 - 0.0874i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.996 - 0.0874i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(571\)
Sign: $0.996 - 0.0874i$
Analytic conductor: \(61.3624\)
Root analytic conductor: \(61.3624\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{571} (389, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 571,\ (1:\ ),\ 0.996 - 0.0874i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.028935325 - 0.1327418568i\)
\(L(\frac12)\) \(\approx\) \(3.028935325 - 0.1327418568i\)
\(L(1)\) \(\approx\) \(1.602933904 + 0.1652409202i\)
\(L(1)\) \(\approx\) \(1.602933904 + 0.1652409202i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad571 \( 1 \)
good2 \( 1 + (0.986 + 0.164i)T \)
3 \( 1 + (-0.991 + 0.131i)T \)
5 \( 1 + (0.746 + 0.665i)T \)
7 \( 1 + (-0.652 - 0.757i)T \)
11 \( 1 + (-0.627 + 0.778i)T \)
13 \( 1 + (-0.846 - 0.533i)T \)
17 \( 1 + (0.340 - 0.940i)T \)
19 \( 1 + (0.724 - 0.689i)T \)
23 \( 1 + (0.431 - 0.901i)T \)
29 \( 1 + (0.245 - 0.969i)T \)
31 \( 1 + (0.789 + 0.614i)T \)
37 \( 1 + (0.997 + 0.0660i)T \)
41 \( 1 + (0.401 - 0.915i)T \)
43 \( 1 + (-0.934 + 0.355i)T \)
47 \( 1 + (-0.789 - 0.614i)T \)
53 \( 1 + (0.461 + 0.887i)T \)
59 \( 1 + (-0.0825 - 0.996i)T \)
61 \( 1 + (0.894 - 0.446i)T \)
67 \( 1 + (0.461 + 0.887i)T \)
71 \( 1 + (0.809 - 0.587i)T \)
73 \( 1 + (-0.997 - 0.0660i)T \)
79 \( 1 + (0.574 - 0.818i)T \)
83 \( 1 + (-0.999 - 0.0330i)T \)
89 \( 1 + (0.999 + 0.0330i)T \)
97 \( 1 + (0.601 - 0.799i)T \)
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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

−23.00927815954978216273029057984, −22.0637144869986010546797618577, −21.54770358584191737168279458494, −21.10849199674610866037634139744, −19.74070892574076298160581517744, −18.96440658789729120597141586300, −18.03995871285719740734974779143, −16.7829824515314061027747471095, −16.43678729808310258665713207168, −15.56804204861822940596830411944, −14.492128325390657439308342848280, −13.33833237869690382455292538444, −12.873876202544633749785230729514, −12.08777448612803116150051356881, −11.376255298299918525174569552260, −10.171369460584124968888561070487, −9.617167504003941361801634384013, −8.08186258002106425013974387261, −6.808216176748138659240042475715, −5.88404771405257143258207411116, −5.49875915063125823919662118031, −4.60902402645005010900451930945, −3.25165546763163699095041323212, −2.07205318971713717336508682987, −1.01249458911725458985115662240, 0.68994446653608850382994178375, 2.35706362722271728046160788959, 3.2044400461106634481316303631, 4.6230310137426468750338749690, 5.17773451195152494328092226656, 6.24942549927052768448700863163, 7.0015432012823289923647111663, 7.53530391086707019690699213708, 9.80361489495932559051370050744, 10.15512539977175600831930400100, 11.081247960997325636103549463784, 12.040316304653370324008278837679, 12.92826988130228536930367301221, 13.53137649981688895494502831477, 14.548601475347512853427722279159, 15.46045958356413302128647223546, 16.20406694413109857994664867026, 17.15066133878381309635150666026, 17.69406824793679166578774655901, 18.748570326215839993113462648660, 20.002591743946472361040700644309, 20.78797117713087835551797313499, 21.64961993722430615931855011750, 22.476871770885522668868295411036, 22.858964673379294420684389619327

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