Properties

Label 1-552-552.5-r0-0-0
Degree $1$
Conductor $552$
Sign $0.987 + 0.159i$
Analytic cond. $2.56347$
Root an. cond. $2.56347$
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  + (0.959 + 0.281i)5-s + (0.654 − 0.755i)7-s + (−0.841 + 0.540i)11-s + (0.654 + 0.755i)13-s + (0.415 − 0.909i)17-s + (0.415 + 0.909i)19-s + (0.841 + 0.540i)25-s + (0.415 − 0.909i)29-s + (−0.142 + 0.989i)31-s + (0.841 − 0.540i)35-s + (−0.959 + 0.281i)37-s + (0.959 + 0.281i)41-s + (−0.142 − 0.989i)43-s − 47-s + (−0.142 − 0.989i)49-s + ⋯
L(s)  = 1  + (0.959 + 0.281i)5-s + (0.654 − 0.755i)7-s + (−0.841 + 0.540i)11-s + (0.654 + 0.755i)13-s + (0.415 − 0.909i)17-s + (0.415 + 0.909i)19-s + (0.841 + 0.540i)25-s + (0.415 − 0.909i)29-s + (−0.142 + 0.989i)31-s + (0.841 − 0.540i)35-s + (−0.959 + 0.281i)37-s + (0.959 + 0.281i)41-s + (−0.142 − 0.989i)43-s − 47-s + (−0.142 − 0.989i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.987 + 0.159i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.987 + 0.159i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(552\)    =    \(2^{3} \cdot 3 \cdot 23\)
Sign: $0.987 + 0.159i$
Analytic conductor: \(2.56347\)
Root analytic conductor: \(2.56347\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{552} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 552,\ (0:\ ),\ 0.987 + 0.159i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.774878345 + 0.1424843133i\)
\(L(\frac12)\) \(\approx\) \(1.774878345 + 0.1424843133i\)
\(L(1)\) \(\approx\) \(1.340640618 + 0.04653056167i\)
\(L(1)\) \(\approx\) \(1.340640618 + 0.04653056167i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
23 \( 1 \)
good5 \( 1 + (0.959 + 0.281i)T \)
7 \( 1 + (0.654 - 0.755i)T \)
11 \( 1 + (-0.841 + 0.540i)T \)
13 \( 1 + (0.654 + 0.755i)T \)
17 \( 1 + (0.415 - 0.909i)T \)
19 \( 1 + (0.415 + 0.909i)T \)
29 \( 1 + (0.415 - 0.909i)T \)
31 \( 1 + (-0.142 + 0.989i)T \)
37 \( 1 + (-0.959 + 0.281i)T \)
41 \( 1 + (0.959 + 0.281i)T \)
43 \( 1 + (-0.142 - 0.989i)T \)
47 \( 1 - T \)
53 \( 1 + (0.654 - 0.755i)T \)
59 \( 1 + (-0.654 - 0.755i)T \)
61 \( 1 + (-0.142 + 0.989i)T \)
67 \( 1 + (0.841 + 0.540i)T \)
71 \( 1 + (-0.841 - 0.540i)T \)
73 \( 1 + (0.415 + 0.909i)T \)
79 \( 1 + (0.654 + 0.755i)T \)
83 \( 1 + (0.959 - 0.281i)T \)
89 \( 1 + (-0.142 - 0.989i)T \)
97 \( 1 + (0.959 + 0.281i)T \)
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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

−23.45293236360691846486714960873, −22.30870782490140064602025327309, −21.4787658921976267968261309602, −21.0686633346135034538084567478, −20.164234609144947972365645675839, −19.00261701634708236108567768011, −18.05598408215234691609934109242, −17.736024601528770591256270248191, −16.603703229769252738548427455224, −15.695069978244754337060048421938, −14.87254545917987771277713641225, −13.88289510327689889599496191344, −13.09935741089253842611738549471, −12.369299585132711177221240606314, −11.09014721464146306701958888925, −10.491181256144008982902670049705, −9.327463864676616257528708193957, −8.53340782457400832692851796212, −7.75891731184763997409173502894, −6.24554070162380347792771865094, −5.563419445870955722439823135441, −4.84928352532703996400812080158, −3.252103868926953572612210482992, −2.28505688099654411429286779652, −1.1527385236807619193131676730, 1.282641554537415290545308041175, 2.225240889169130845939615729, 3.484139420440316945792349705356, 4.72692603372414011322288679436, 5.52512150352933574245580945283, 6.6872697320268713228384343450, 7.49217823881895253208645636238, 8.51958689988169492130582208194, 9.72276237539017434883125728111, 10.30657240771711285364476241030, 11.208269949691633425241710030273, 12.225569838609383171270914137918, 13.42191350070419740173635145500, 13.94830231978911139420613533271, 14.64767604235253384424804393534, 15.89110402218137427016844603682, 16.673130026181992257356073509299, 17.659984790442111489122189634145, 18.18112670838006031919013170479, 19.0115864230450299178602355830, 20.353801208316391500481268903307, 20.942140921517195804049201899193, 21.41488489379251434415877294719, 22.76759171718482152837860821747, 23.202022151266620697295737809150

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