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Properties

Label 1-571-571.150-r0-0-0
Degree $1$
Conductor $571$
Sign $-0.653 + 0.756i$
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  + (0.975 + 0.218i)2-s + (−0.926 + 0.376i)3-s + (0.904 + 0.426i)4-s + (0.851 + 0.523i)5-s + (−0.986 + 0.164i)6-s + (−0.401 + 0.915i)7-s + (0.789 + 0.614i)8-s + (0.716 − 0.697i)9-s + (0.716 + 0.697i)10-s + (−0.962 − 0.272i)11-s + (−0.998 − 0.0550i)12-s + (−0.754 + 0.656i)13-s + (−0.592 + 0.805i)14-s + (−0.986 − 0.164i)15-s + (0.635 + 0.771i)16-s + (0.635 + 0.771i)17-s + ⋯
L(s)  = 1  + (0.975 + 0.218i)2-s + (−0.926 + 0.376i)3-s + (0.904 + 0.426i)4-s + (0.851 + 0.523i)5-s + (−0.986 + 0.164i)6-s + (−0.401 + 0.915i)7-s + (0.789 + 0.614i)8-s + (0.716 − 0.697i)9-s + (0.716 + 0.697i)10-s + (−0.962 − 0.272i)11-s + (−0.998 − 0.0550i)12-s + (−0.754 + 0.656i)13-s + (−0.592 + 0.805i)14-s + (−0.986 − 0.164i)15-s + (0.635 + 0.771i)16-s + (0.635 + 0.771i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.653 + 0.756i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.653 + 0.756i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(571\)
Sign: $-0.653 + 0.756i$
Analytic conductor: \(2.65171\)
Root analytic conductor: \(2.65171\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{571} (150, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 571,\ (0:\ ),\ -0.653 + 0.756i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7607993719 + 1.663127487i\)
\(L(\frac12)\) \(\approx\) \(0.7607993719 + 1.663127487i\)
\(L(1)\) \(\approx\) \(1.222844178 + 0.8088958734i\)
\(L(1)\) \(\approx\) \(1.222844178 + 0.8088958734i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad571 \( 1 \)
good2 \( 1 + (0.975 + 0.218i)T \)
3 \( 1 + (-0.926 + 0.376i)T \)
5 \( 1 + (0.851 + 0.523i)T \)
7 \( 1 + (-0.401 + 0.915i)T \)
11 \( 1 + (-0.962 - 0.272i)T \)
13 \( 1 + (-0.754 + 0.656i)T \)
17 \( 1 + (0.635 + 0.771i)T \)
19 \( 1 + (-0.926 + 0.376i)T \)
23 \( 1 + (0.789 - 0.614i)T \)
29 \( 1 + (-0.191 - 0.981i)T \)
31 \( 1 + (-0.986 + 0.164i)T \)
37 \( 1 + (-0.754 - 0.656i)T \)
41 \( 1 + (0.851 + 0.523i)T \)
43 \( 1 + (0.716 + 0.697i)T \)
47 \( 1 + (0.350 - 0.936i)T \)
53 \( 1 + (0.975 - 0.218i)T \)
59 \( 1 + (-0.401 + 0.915i)T \)
61 \( 1 + (-0.298 - 0.954i)T \)
67 \( 1 + (-0.298 + 0.954i)T \)
71 \( 1 + (-0.5 + 0.866i)T \)
73 \( 1 + (-0.191 + 0.981i)T \)
79 \( 1 + (-0.821 - 0.569i)T \)
83 \( 1 + (0.350 - 0.936i)T \)
89 \( 1 + (0.635 + 0.771i)T \)
97 \( 1 + (0.904 + 0.426i)T \)
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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

−22.87895474584037030359052996240, −22.31114955367769285551548571840, −21.341644208169951924752923820220, −20.70492361261734087576358081684, −19.822626740573919727778357559728, −18.84484151420614249387120382927, −17.72897345012450254373797720176, −16.9710866387261335250311936449, −16.32651116920431329402861328176, −15.384662450758994041810917538472, −14.16850241839925275246522694933, −13.307813456675905161413080051857, −12.81543036275355559351306188142, −12.196022638408357099935043615316, −10.77070718700965063611623370537, −10.47635180454841761926712946147, −9.458551897318194696971770913743, −7.53776607781922228625702869430, −7.0540557939001982090089657295, −5.871079328036282587319512667182, −5.19983730730567307047864127744, −4.53742257935926787888733901025, −3.03694725033608458655800487551, −1.920376016027930267590175528789, −0.72395658467903274622393176926, 1.954275526076548873029233120141, 2.81339196814070406906616238505, 4.04837940453381699621502833958, 5.2163224625478660634775642168, 5.792130850886212099052772197110, 6.44881769965408765861726427395, 7.43313251809613560465191175217, 8.92476553034998065362832227281, 10.12670566849711131241331242119, 10.72388331341320614655859779384, 11.70875790718370499869999069028, 12.662953243459267606044702360275, 13.08319986665194121280095584053, 14.52800869491771267414753623311, 14.95185076657077280995025497034, 15.95607122308022537249779030876, 16.73026816440782364379375670692, 17.41589207902293489131160381193, 18.54181953554619533881206485573, 19.21729675605178907363676745488, 20.87782943246955750845918530563, 21.51905446660202077034698487486, 21.69066974734643355357489266008, 22.78009128539709732251537767664, 23.23680003175824055740172343260

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