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Properties

Label 1-572-572.515-r0-0-0
Degree $1$
Conductor $572$
Sign $0.768 + 0.639i$
Analytic cond. $2.65635$
Root an. cond. $2.65635$
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  + (−0.309 + 0.951i)3-s + (−0.587 − 0.809i)5-s + (−0.951 + 0.309i)7-s + (−0.809 − 0.587i)9-s + (0.951 − 0.309i)15-s + (0.809 − 0.587i)17-s + (−0.951 − 0.309i)19-s − i·21-s + 23-s + (−0.309 + 0.951i)25-s + (0.809 − 0.587i)27-s + (0.309 + 0.951i)29-s + (−0.587 + 0.809i)31-s + (0.809 + 0.587i)35-s + (0.951 − 0.309i)37-s + ⋯
L(s)  = 1  + (−0.309 + 0.951i)3-s + (−0.587 − 0.809i)5-s + (−0.951 + 0.309i)7-s + (−0.809 − 0.587i)9-s + (0.951 − 0.309i)15-s + (0.809 − 0.587i)17-s + (−0.951 − 0.309i)19-s − i·21-s + 23-s + (−0.309 + 0.951i)25-s + (0.809 − 0.587i)27-s + (0.309 + 0.951i)29-s + (−0.587 + 0.809i)31-s + (0.809 + 0.587i)35-s + (0.951 − 0.309i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 572 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.768 + 0.639i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 572 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.768 + 0.639i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(572\)    =    \(2^{2} \cdot 11 \cdot 13\)
Sign: $0.768 + 0.639i$
Analytic conductor: \(2.65635\)
Root analytic conductor: \(2.65635\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{572} (515, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 572,\ (0:\ ),\ 0.768 + 0.639i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7938458478 + 0.2871864124i\)
\(L(\frac12)\) \(\approx\) \(0.7938458478 + 0.2871864124i\)
\(L(1)\) \(\approx\) \(0.7624240746 + 0.1488242567i\)
\(L(1)\) \(\approx\) \(0.7624240746 + 0.1488242567i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
11 \( 1 \)
13 \( 1 \)
good3 \( 1 + (-0.309 + 0.951i)T \)
5 \( 1 + (-0.587 - 0.809i)T \)
7 \( 1 + (-0.951 + 0.309i)T \)
17 \( 1 + (0.809 - 0.587i)T \)
19 \( 1 + (-0.951 - 0.309i)T \)
23 \( 1 + T \)
29 \( 1 + (0.309 + 0.951i)T \)
31 \( 1 + (-0.587 + 0.809i)T \)
37 \( 1 + (0.951 - 0.309i)T \)
41 \( 1 + (0.951 + 0.309i)T \)
43 \( 1 + T \)
47 \( 1 + (0.951 + 0.309i)T \)
53 \( 1 + (-0.809 - 0.587i)T \)
59 \( 1 + (-0.951 + 0.309i)T \)
61 \( 1 + (-0.809 + 0.587i)T \)
67 \( 1 - iT \)
71 \( 1 + (0.587 + 0.809i)T \)
73 \( 1 + (0.951 - 0.309i)T \)
79 \( 1 + (0.809 + 0.587i)T \)
83 \( 1 + (0.587 + 0.809i)T \)
89 \( 1 - iT \)
97 \( 1 + (0.587 - 0.809i)T \)
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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

−23.26256934954023967805574996710, −22.59338083992611975420988559800, −21.73631200524260719083886249511, −20.47790903627532577100615574065, −19.35281911067853379171440937984, −19.13126563852807918859479871747, −18.39087721087955059718517485443, −17.21196020351593229437162545190, −16.66669615778919012131596108179, −15.52104164604398180210732475950, −14.64719159721626036765482855344, −13.74760520128662634497551716212, −12.79154842585116443659018096782, −12.231917345775571653491054451881, −11.10198262717727827881356457097, −10.517910621907615628728162260065, −9.311599089363232041120869902783, −7.98848792535369032338520912210, −7.41672379276406808505568357434, −6.41693597018943871314499568622, −5.90727170481869403696194270404, −4.25642461032578157312553459572, −3.20928976217304984285054821852, −2.27150538711568672286992859941, −0.69078693938273177085103121279, 0.80612546191416214547204995878, 2.79903957892994105067006285726, 3.67947310183303611597912823401, 4.67043043213732372345626209152, 5.459735606127150833509063450030, 6.49057826765189883531350980009, 7.71446121573554478974387175091, 9.076139297174989133175506599049, 9.1943733027315887770019230023, 10.47432466182603743832739429760, 11.26758488722819570258339718832, 12.3717504598825906889082126398, 12.78869226425183430672461558411, 14.18797832548433331556644051468, 15.167953014191239642565114941705, 15.88629672589240574725627438907, 16.50230265918718473776938173656, 17.11247819373572748284728188687, 18.345919595844448465691673757077, 19.43741455230268139044841620056, 19.97466032848959630385951601044, 21.012032558863858194574413835624, 21.536664074819139549236723604852, 22.62313557916569624309892311751, 23.179187514454821539227637866958

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