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Properties

Label 2-552-23.4-c1-0-6
Degree $2$
Conductor $552$
Sign $0.941 - 0.338i$
Analytic cond. $4.40774$
Root an. cond. $2.09946$
Motivic weight $1$
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)3-s + (2.64 + 1.69i)5-s + (0.0583 + 0.405i)7-s + (0.841 − 0.540i)9-s + (0.742 − 1.62i)11-s + (−0.395 + 2.75i)13-s + (3.01 + 0.885i)15-s + (1.07 + 1.24i)17-s + (0.916 − 1.05i)19-s + (0.170 + 0.372i)21-s + (−4.69 − 0.958i)23-s + (2.02 + 4.44i)25-s + (0.654 − 0.755i)27-s + (0.646 + 0.745i)29-s + (0.686 + 0.201i)31-s + ⋯
L(s)  = 1  + (0.553 − 0.162i)3-s + (1.18 + 0.760i)5-s + (0.0220 + 0.153i)7-s + (0.280 − 0.180i)9-s + (0.223 − 0.490i)11-s + (−0.109 + 0.763i)13-s + (0.778 + 0.228i)15-s + (0.260 + 0.301i)17-s + (0.210 − 0.242i)19-s + (0.0371 + 0.0813i)21-s + (−0.979 − 0.199i)23-s + (0.405 + 0.888i)25-s + (0.126 − 0.145i)27-s + (0.119 + 0.138i)29-s + (0.123 + 0.0361i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.941 - 0.338i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.941 - 0.338i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(552\)    =    \(2^{3} \cdot 3 \cdot 23\)
Sign: $0.941 - 0.338i$
Analytic conductor: \(4.40774\)
Root analytic conductor: \(2.09946\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{552} (73, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 552,\ (\ :1/2),\ 0.941 - 0.338i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.08317 + 0.362973i\)
\(L(\frac12)\) \(\approx\) \(2.08317 + 0.362973i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-0.959 + 0.281i)T \)
23 \( 1 + (4.69 + 0.958i)T \)
good5 \( 1 + (-2.64 - 1.69i)T + (2.07 + 4.54i)T^{2} \)
7 \( 1 + (-0.0583 - 0.405i)T + (-6.71 + 1.97i)T^{2} \)
11 \( 1 + (-0.742 + 1.62i)T + (-7.20 - 8.31i)T^{2} \)
13 \( 1 + (0.395 - 2.75i)T + (-12.4 - 3.66i)T^{2} \)
17 \( 1 + (-1.07 - 1.24i)T + (-2.41 + 16.8i)T^{2} \)
19 \( 1 + (-0.916 + 1.05i)T + (-2.70 - 18.8i)T^{2} \)
29 \( 1 + (-0.646 - 0.745i)T + (-4.12 + 28.7i)T^{2} \)
31 \( 1 + (-0.686 - 0.201i)T + (26.0 + 16.7i)T^{2} \)
37 \( 1 + (2.07 - 1.33i)T + (15.3 - 33.6i)T^{2} \)
41 \( 1 + (6.15 + 3.95i)T + (17.0 + 37.2i)T^{2} \)
43 \( 1 + (-7.39 + 2.17i)T + (36.1 - 23.2i)T^{2} \)
47 \( 1 + 2.85T + 47T^{2} \)
53 \( 1 + (-0.372 - 2.59i)T + (-50.8 + 14.9i)T^{2} \)
59 \( 1 + (-1.19 + 8.32i)T + (-56.6 - 16.6i)T^{2} \)
61 \( 1 + (-7.71 - 2.26i)T + (51.3 + 32.9i)T^{2} \)
67 \( 1 + (4.60 + 10.0i)T + (-43.8 + 50.6i)T^{2} \)
71 \( 1 + (0.0976 + 0.213i)T + (-46.4 + 53.6i)T^{2} \)
73 \( 1 + (4.37 - 5.05i)T + (-10.3 - 72.2i)T^{2} \)
79 \( 1 + (-1.17 + 8.13i)T + (-75.7 - 22.2i)T^{2} \)
83 \( 1 + (13.0 - 8.37i)T + (34.4 - 75.4i)T^{2} \)
89 \( 1 + (12.6 - 3.72i)T + (74.8 - 48.1i)T^{2} \)
97 \( 1 + (-6.89 - 4.43i)T + (40.2 + 88.2i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.63477266214996521241986040155, −9.920498120544555058054864651278, −9.150493931176398829742184717155, −8.285521660196531060146205704096, −7.08426418198738576677259424005, −6.35678969260970590808384363049, −5.46057023353817014146932301136, −3.95817038271867346484416898920, −2.74555213590200566630029181949, −1.76734732941506879183987981933, 1.42327200833309758053055455239, 2.64628039342493105796688778519, 4.07532965693483236326141314818, 5.19102226737416847067143128124, 5.96978489368664126191985664058, 7.24498939731569882885616956593, 8.212285232187454458729661636280, 9.077907540986827264750828483739, 9.888795173537655399124984468774, 10.29300355967371141380956640632

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