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Properties

Label 1-571-571.128-r1-0-0
Degree $1$
Conductor $571$
Sign $0.991 + 0.127i$
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  + (−0.451 − 0.892i)2-s + (−0.350 − 0.936i)3-s + (−0.592 + 0.805i)4-s + (−0.926 + 0.376i)5-s + (−0.677 + 0.735i)6-s + (0.879 + 0.475i)7-s + (0.986 + 0.164i)8-s + (−0.754 + 0.656i)9-s + (0.754 + 0.656i)10-s + (−0.191 − 0.981i)11-s + (0.962 + 0.272i)12-s + (0.904 − 0.426i)13-s + (0.0275 − 0.999i)14-s + (0.677 + 0.735i)15-s + (−0.298 − 0.954i)16-s + (0.298 + 0.954i)17-s + ⋯
L(s)  = 1  + (−0.451 − 0.892i)2-s + (−0.350 − 0.936i)3-s + (−0.592 + 0.805i)4-s + (−0.926 + 0.376i)5-s + (−0.677 + 0.735i)6-s + (0.879 + 0.475i)7-s + (0.986 + 0.164i)8-s + (−0.754 + 0.656i)9-s + (0.754 + 0.656i)10-s + (−0.191 − 0.981i)11-s + (0.962 + 0.272i)12-s + (0.904 − 0.426i)13-s + (0.0275 − 0.999i)14-s + (0.677 + 0.735i)15-s + (−0.298 − 0.954i)16-s + (0.298 + 0.954i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.991 + 0.127i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.991 + 0.127i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(571\)
Sign: $0.991 + 0.127i$
Analytic conductor: \(61.3624\)
Root analytic conductor: \(61.3624\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{571} (128, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 571,\ (1:\ ),\ 0.991 + 0.127i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5882994836 + 0.03759798231i\)
\(L(\frac12)\) \(\approx\) \(0.5882994836 + 0.03759798231i\)
\(L(1)\) \(\approx\) \(0.5321522960 - 0.3038417173i\)
\(L(1)\) \(\approx\) \(0.5321522960 - 0.3038417173i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad571 \( 1 \)
good2 \( 1 + (-0.451 - 0.892i)T \)
3 \( 1 + (-0.350 - 0.936i)T \)
5 \( 1 + (-0.926 + 0.376i)T \)
7 \( 1 + (0.879 + 0.475i)T \)
11 \( 1 + (-0.191 - 0.981i)T \)
13 \( 1 + (0.904 - 0.426i)T \)
17 \( 1 + (0.298 + 0.954i)T \)
19 \( 1 + (-0.350 - 0.936i)T \)
23 \( 1 + (-0.986 + 0.164i)T \)
29 \( 1 + (-0.821 - 0.569i)T \)
31 \( 1 + (-0.677 + 0.735i)T \)
37 \( 1 + (0.904 + 0.426i)T \)
41 \( 1 + (0.926 - 0.376i)T \)
43 \( 1 + (-0.754 - 0.656i)T \)
47 \( 1 + (-0.975 - 0.218i)T \)
53 \( 1 + (-0.451 + 0.892i)T \)
59 \( 1 + (-0.879 - 0.475i)T \)
61 \( 1 + (-0.998 - 0.0550i)T \)
67 \( 1 + (0.998 - 0.0550i)T \)
71 \( 1 + (0.5 + 0.866i)T \)
73 \( 1 + (0.821 - 0.569i)T \)
79 \( 1 + (-0.993 + 0.110i)T \)
83 \( 1 + (0.975 + 0.218i)T \)
89 \( 1 + (0.298 + 0.954i)T \)
97 \( 1 + (-0.592 + 0.805i)T \)
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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

−23.12161543648325807812194026909, −22.65628250460505125381443916042, −21.24122882403553007178226409669, −20.41629913925953860232001873409, −19.91712287463716660223280000785, −18.43479903176428341152097694135, −17.99687138619349714757340757150, −16.74106467159742080381919879940, −16.43934329708838047655922237655, −15.59389603467301448861256309635, −14.7541571794729997296511054307, −14.23075423364497385708003939014, −12.82261056038716802371854918309, −11.56599366680483363137498589974, −10.95755894929795710393748371634, −9.91905254180251443141981231382, −9.12332848942286090287972899696, −8.0824822200538737879078855822, −7.52570110266105727132829266071, −6.25874244019350493024950150291, −5.15885984226366709506577861723, −4.43337958742874746985027555811, −3.78641527766637722752044894424, −1.571921169086903155626165630787, −0.24956761626686759590958090451, 0.83411588577183189286518265376, 1.90145367891573813792325808714, 2.997642144627506518998341796191, 3.989988523796010425315513037407, 5.31107108276072559595311188329, 6.4061385113898033864091979969, 7.85435291051392842170375401709, 8.06922582717337424329101187115, 8.942395468041290900106206957404, 10.7402057862823344338610192952, 11.04467946389522666302405945059, 11.77724827703436358956446192929, 12.59975744111497837206464120046, 13.45333993260807089236239220657, 14.37176201056725953729007955061, 15.55301658299087375647539579830, 16.61137062504194959238065058596, 17.54061429942932789273018123291, 18.38025709098490634138462026279, 18.729406848069545933577519859767, 19.61906147360883061814775146509, 20.28218222473493043516237327084, 21.50426692977420070455282942787, 22.03459337221539939945653433563, 23.16794358300554664475925873289

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