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Properties

Label 1-552-552.371-r0-0-0
Degree $1$
Conductor $552$
Sign $0.381 - 0.924i$
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  + (−0.142 − 0.989i)5-s + (−0.415 + 0.909i)7-s + (0.959 + 0.281i)11-s + (−0.415 − 0.909i)13-s + (−0.841 − 0.540i)17-s + (0.841 − 0.540i)19-s + (−0.959 + 0.281i)25-s + (0.841 + 0.540i)29-s + (0.654 − 0.755i)31-s + (0.959 + 0.281i)35-s + (0.142 − 0.989i)37-s + (0.142 + 0.989i)41-s + (−0.654 − 0.755i)43-s + 47-s + (−0.654 − 0.755i)49-s + ⋯
L(s)  = 1  + (−0.142 − 0.989i)5-s + (−0.415 + 0.909i)7-s + (0.959 + 0.281i)11-s + (−0.415 − 0.909i)13-s + (−0.841 − 0.540i)17-s + (0.841 − 0.540i)19-s + (−0.959 + 0.281i)25-s + (0.841 + 0.540i)29-s + (0.654 − 0.755i)31-s + (0.959 + 0.281i)35-s + (0.142 − 0.989i)37-s + (0.142 + 0.989i)41-s + (−0.654 − 0.755i)43-s + 47-s + (−0.654 − 0.755i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.381 - 0.924i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.381 - 0.924i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(552\)    =    \(2^{3} \cdot 3 \cdot 23\)
Sign: $0.381 - 0.924i$
Analytic conductor: \(2.56347\)
Root analytic conductor: \(2.56347\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{552} (371, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 552,\ (0:\ ),\ 0.381 - 0.924i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9867414466 - 0.6600880348i\)
\(L(\frac12)\) \(\approx\) \(0.9867414466 - 0.6600880348i\)
\(L(1)\) \(\approx\) \(0.9847568859 - 0.2202138332i\)
\(L(1)\) \(\approx\) \(0.9847568859 - 0.2202138332i\)

Euler product

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

−23.4296017620545538131023009650, −22.57587674499107107554702374589, −22.052470538081459602296128496197, −21.11296775480587672490494768982, −19.84763973225483007700719754720, −19.47494833034871389477436915133, −18.5917382447096836331713749209, −17.54475597097475563603038661651, −16.84931131181701281528366370275, −15.92615515957998108184986020891, −14.95134001468166399219888421463, −13.99095528521474426535689375887, −13.67050294957436375506864364765, −12.1756869742112014549526138718, −11.48433453247732409992308744954, −10.502649766177375880466865892893, −9.819459322834245430843068711981, −8.74014232928109585165309380149, −7.50139667294418631290554443945, −6.75198957671148241674683562183, −6.13454361952994277955180929615, −4.45186446071675539400857482273, −3.73908606857600334212421649432, −2.69097180201765354316363123948, −1.29723035435667914515131094338, 0.68740622637951511566018847800, 2.11556713843630199235974235861, 3.24243087053389083022929131622, 4.53063922499848081230731328871, 5.29896933048978462789507714143, 6.31305035090970381675829454104, 7.41952558974493509449436046563, 8.55661185484181415480076903610, 9.207259365528007202132046898154, 9.94842030869353294982368462804, 11.43157925802801177430052792190, 12.08255862000468021224893654660, 12.83017549741532009205777457257, 13.67796007988202807895102127269, 14.88904216052944197318980792243, 15.66406154722263892443688074135, 16.31721305426472761277611304394, 17.40826143087562072326178159372, 17.979828560737784085132108039335, 19.20195541448793358731224893884, 19.95508537664392045583899957016, 20.46202705671783866639711192132, 21.713515342136349400506572976679, 22.26225701263157595802317889548, 23.11274552742731385510708011533

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