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Properties

Label 1-579-579.8-r1-0-0
Degree $1$
Conductor $579$
Sign $0.418 - 0.908i$
Analytic cond. $62.2221$
Root an. cond. $62.2221$
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  + (−0.923 − 0.382i)2-s + (0.707 + 0.707i)4-s + (0.980 + 0.195i)5-s + i·7-s + (−0.382 − 0.923i)8-s + (−0.831 − 0.555i)10-s + (−0.195 − 0.980i)11-s + (0.831 − 0.555i)13-s + (0.382 − 0.923i)14-s + i·16-s + (0.980 − 0.195i)17-s + (0.980 + 0.195i)19-s + (0.555 + 0.831i)20-s + (−0.195 + 0.980i)22-s + (−0.923 − 0.382i)23-s + ⋯
L(s)  = 1  + (−0.923 − 0.382i)2-s + (0.707 + 0.707i)4-s + (0.980 + 0.195i)5-s + i·7-s + (−0.382 − 0.923i)8-s + (−0.831 − 0.555i)10-s + (−0.195 − 0.980i)11-s + (0.831 − 0.555i)13-s + (0.382 − 0.923i)14-s + i·16-s + (0.980 − 0.195i)17-s + (0.980 + 0.195i)19-s + (0.555 + 0.831i)20-s + (−0.195 + 0.980i)22-s + (−0.923 − 0.382i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 579 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.418 - 0.908i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 579 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.418 - 0.908i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(579\)    =    \(3 \cdot 193\)
Sign: $0.418 - 0.908i$
Analytic conductor: \(62.2221\)
Root analytic conductor: \(62.2221\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{579} (8, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 579,\ (1:\ ),\ 0.418 - 0.908i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.377407948 - 0.8814384339i\)
\(L(\frac12)\) \(\approx\) \(1.377407948 - 0.8814384339i\)
\(L(1)\) \(\approx\) \(0.9145625879 - 0.1809009446i\)
\(L(1)\) \(\approx\) \(0.9145625879 - 0.1809009446i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
193 \( 1 \)
good2 \( 1 + (-0.923 - 0.382i)T \)
5 \( 1 + (0.980 + 0.195i)T \)
7 \( 1 + iT \)
11 \( 1 + (-0.195 - 0.980i)T \)
13 \( 1 + (0.831 - 0.555i)T \)
17 \( 1 + (0.980 - 0.195i)T \)
19 \( 1 + (0.980 + 0.195i)T \)
23 \( 1 + (-0.923 - 0.382i)T \)
29 \( 1 + (0.555 - 0.831i)T \)
31 \( 1 + (-0.923 - 0.382i)T \)
37 \( 1 + (-0.555 - 0.831i)T \)
41 \( 1 + (-0.195 - 0.980i)T \)
43 \( 1 - iT \)
47 \( 1 + (0.831 - 0.555i)T \)
53 \( 1 + (-0.831 + 0.555i)T \)
59 \( 1 + (0.707 + 0.707i)T \)
61 \( 1 + (0.980 - 0.195i)T \)
67 \( 1 + (-0.923 - 0.382i)T \)
71 \( 1 + (-0.980 - 0.195i)T \)
73 \( 1 + (-0.555 + 0.831i)T \)
79 \( 1 + (-0.195 - 0.980i)T \)
83 \( 1 + (-0.382 + 0.923i)T \)
89 \( 1 + (-0.831 - 0.555i)T \)
97 \( 1 + (0.923 - 0.382i)T \)
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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

−23.58733989746983933274944702498, −22.37967974813915333177652594588, −21.18087991474118105287505273008, −20.48460097490779521795051191046, −19.93974807022661092520192048732, −18.7648532078359539047995243945, −17.89243582133498124424500970886, −17.50089766444929382888123318942, −16.42665631263859953225862354441, −16.04113190324404224050163890138, −14.60434973913667765082806663476, −14.06912914036826908867328395530, −13.097663207163151010753333888834, −11.9020604110205916037041057102, −10.80726368314471951879569483253, −9.98795522374572088748901263877, −9.54307129221801112503970589503, −8.400103286807998449700476253538, −7.4225926830389627049729184461, −6.66794647781534756707080725564, −5.6953608590453309950140603276, −4.70407662186174924820050731845, −3.18826994982083139502053643507, −1.692516844372863778779195783458, −1.15534333894155734393619592808, 0.58843784568655415122137739301, 1.77364319403837591500384268035, 2.74773073132991272881626096638, 3.57355533040178741552887608329, 5.71818708729082123256259312554, 5.836153156514362079699180906727, 7.2839245449636548071837195591, 8.36152052623441550780947582682, 8.988768018937483535328575951, 9.94457808321741814815344667772, 10.62507342954367941662515368818, 11.63881770144447683662854260022, 12.42171156419481159707912976838, 13.45403259090117754005978536992, 14.32114583667264505069833939082, 15.62681934903677612996035362076, 16.19614368938555238569464775237, 17.16810650976924157475164043275, 18.16086068821579129094628803652, 18.47299866923788823260429275742, 19.24890403503215184147012437710, 20.58132541434838227987450913889, 20.97653548258770562055935655283, 21.90726405643990908163489614334, 22.419218519035386978640188022830

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