Properties

Label 1-571-571.400-r0-0-0
Degree $1$
Conductor $571$
Sign $-0.924 - 0.380i$
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  + (−0.677 + 0.735i)2-s + (−0.340 + 0.940i)3-s + (−0.0825 − 0.996i)4-s + (0.431 − 0.901i)5-s + (−0.461 − 0.887i)6-s + (−0.995 − 0.0990i)7-s + (0.789 + 0.614i)8-s + (−0.768 − 0.639i)9-s + (0.371 + 0.928i)10-s + (0.371 − 0.928i)11-s + (0.965 + 0.261i)12-s + (−0.956 + 0.293i)13-s + (0.746 − 0.665i)14-s + (0.701 + 0.712i)15-s + (−0.986 + 0.164i)16-s + (0.894 + 0.446i)17-s + ⋯
L(s)  = 1  + (−0.677 + 0.735i)2-s + (−0.340 + 0.940i)3-s + (−0.0825 − 0.996i)4-s + (0.431 − 0.901i)5-s + (−0.461 − 0.887i)6-s + (−0.995 − 0.0990i)7-s + (0.789 + 0.614i)8-s + (−0.768 − 0.639i)9-s + (0.371 + 0.928i)10-s + (0.371 − 0.928i)11-s + (0.965 + 0.261i)12-s + (−0.956 + 0.293i)13-s + (0.746 − 0.665i)14-s + (0.701 + 0.712i)15-s + (−0.986 + 0.164i)16-s + (0.894 + 0.446i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.924 - 0.380i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.924 - 0.380i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(571\)
Sign: $-0.924 - 0.380i$
Analytic conductor: \(2.65171\)
Root analytic conductor: \(2.65171\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{571} (400, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 571,\ (0:\ ),\ -0.924 - 0.380i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.02933354135 + 0.1484147742i\)
\(L(\frac12)\) \(\approx\) \(0.02933354135 + 0.1484147742i\)
\(L(1)\) \(\approx\) \(0.4659492173 + 0.2041796274i\)
\(L(1)\) \(\approx\) \(0.4659492173 + 0.2041796274i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad571 \( 1 \)
good2 \( 1 + (-0.677 + 0.735i)T \)
3 \( 1 + (-0.340 + 0.940i)T \)
5 \( 1 + (0.431 - 0.901i)T \)
7 \( 1 + (-0.995 - 0.0990i)T \)
11 \( 1 + (0.371 - 0.928i)T \)
13 \( 1 + (-0.956 + 0.293i)T \)
17 \( 1 + (0.894 + 0.446i)T \)
19 \( 1 + (-0.999 - 0.0330i)T \)
23 \( 1 + (0.828 + 0.560i)T \)
29 \( 1 + (0.945 + 0.324i)T \)
31 \( 1 + (-0.986 + 0.164i)T \)
37 \( 1 + (-0.574 + 0.818i)T \)
41 \( 1 + (-0.879 + 0.475i)T \)
43 \( 1 + (-0.846 - 0.533i)T \)
47 \( 1 + (-0.986 + 0.164i)T \)
53 \( 1 + (0.115 + 0.993i)T \)
59 \( 1 + (-0.401 + 0.915i)T \)
61 \( 1 + (-0.909 - 0.416i)T \)
67 \( 1 + (0.115 + 0.993i)T \)
71 \( 1 + (0.309 + 0.951i)T \)
73 \( 1 + (-0.574 + 0.818i)T \)
79 \( 1 + (-0.973 + 0.229i)T \)
83 \( 1 + (-0.461 - 0.887i)T \)
89 \( 1 + (-0.461 - 0.887i)T \)
97 \( 1 + (0.652 + 0.757i)T \)
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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

−22.741293666710713883314724707204, −22.06671820668750299864442945501, −21.10906428009057039145310289934, −19.85927247795770844280520965259, −19.36816070647814821301684539420, −18.6549962466705657937739610554, −17.953091930241421707308472485154, −17.150389159271503429552489917649, −16.56689511188558492444237012770, −15.08699171836230493387547113167, −14.12616403230455451129907920999, −13.02496109539286328664837858359, −12.48745328934990517220151924800, −11.73712548389575382343269367211, −10.59793166730776342205280738234, −9.99178038320163178570022502236, −9.1030899058680421075690476498, −7.80330557557031929586787502759, −6.97888978791499093233884654240, −6.477133725828691256149191188368, −5.02564781502806907363063451454, −3.39420169250988715423388025651, −2.562189780257800044751496639155, −1.75798361625972289418780333381, −0.104989450615792243274128709141, 1.31673677331565295934357828577, 3.10421725588180924943304433348, 4.39548922897682776051713946030, 5.330654714531331099549502581597, 6.04005781074536932157593797471, 6.9319179595108270170338192285, 8.46146936394560079241134951743, 8.982231746295653176698599887905, 9.87192052407225529848971852613, 10.35409434855761879493896767888, 11.59137261432260037121843065981, 12.672089598888178260241388858770, 13.7771383097389791859268647645, 14.674993450901642976602874572335, 15.56472597304660876856986964259, 16.45840158904831560165356301410, 16.86782324843681587931479065509, 17.30823464391741938172035890804, 18.732677541248454644833839176509, 19.55134071061467913993816804966, 20.18679538846785931337885820144, 21.4736972581472737668031638247, 21.86832519106525271083904349217, 23.14628295554911929966839343847, 23.666478557570430042265356312335

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