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Properties

Label 2-552-23.9-c1-0-2
Degree $2$
Conductor $552$
Sign $-0.289 - 0.957i$
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  + (0.654 + 0.755i)3-s + (−0.291 − 2.02i)5-s + (−2.05 + 4.50i)7-s + (−0.142 + 0.989i)9-s + (−4.13 − 1.21i)11-s + (2.04 + 4.47i)13-s + (1.34 − 1.54i)15-s + (4.19 + 2.69i)17-s + (−5.89 + 3.79i)19-s + (−4.74 + 1.39i)21-s + (2.12 + 4.29i)23-s + (0.764 − 0.224i)25-s + (−0.841 + 0.540i)27-s + (−2.36 − 1.51i)29-s + (3.39 − 3.91i)31-s + ⋯
L(s)  = 1  + (0.378 + 0.436i)3-s + (−0.130 − 0.907i)5-s + (−0.777 + 1.70i)7-s + (−0.0474 + 0.329i)9-s + (−1.24 − 0.366i)11-s + (0.567 + 1.24i)13-s + (0.346 − 0.400i)15-s + (1.01 + 0.653i)17-s + (−1.35 + 0.869i)19-s + (−1.03 + 0.304i)21-s + (0.443 + 0.896i)23-s + (0.152 − 0.0448i)25-s + (−0.161 + 0.104i)27-s + (−0.438 − 0.281i)29-s + (0.608 − 0.702i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.289 - 0.957i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.289 - 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(552\)    =    \(2^{3} \cdot 3 \cdot 23\)
Sign: $-0.289 - 0.957i$
Analytic conductor: \(4.40774\)
Root analytic conductor: \(2.09946\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{552} (193, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 552,\ (\ :1/2),\ -0.289 - 0.957i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.676073 + 0.911065i\)
\(L(\frac12)\) \(\approx\) \(0.676073 + 0.911065i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.654 - 0.755i)T \)
23 \( 1 + (-2.12 - 4.29i)T \)
good5 \( 1 + (0.291 + 2.02i)T + (-4.79 + 1.40i)T^{2} \)
7 \( 1 + (2.05 - 4.50i)T + (-4.58 - 5.29i)T^{2} \)
11 \( 1 + (4.13 + 1.21i)T + (9.25 + 5.94i)T^{2} \)
13 \( 1 + (-2.04 - 4.47i)T + (-8.51 + 9.82i)T^{2} \)
17 \( 1 + (-4.19 - 2.69i)T + (7.06 + 15.4i)T^{2} \)
19 \( 1 + (5.89 - 3.79i)T + (7.89 - 17.2i)T^{2} \)
29 \( 1 + (2.36 + 1.51i)T + (12.0 + 26.3i)T^{2} \)
31 \( 1 + (-3.39 + 3.91i)T + (-4.41 - 30.6i)T^{2} \)
37 \( 1 + (-0.133 + 0.929i)T + (-35.5 - 10.4i)T^{2} \)
41 \( 1 + (0.307 + 2.13i)T + (-39.3 + 11.5i)T^{2} \)
43 \( 1 + (-1.77 - 2.05i)T + (-6.11 + 42.5i)T^{2} \)
47 \( 1 - 6.81T + 47T^{2} \)
53 \( 1 + (5.84 - 12.7i)T + (-34.7 - 40.0i)T^{2} \)
59 \( 1 + (3.18 + 6.97i)T + (-38.6 + 44.5i)T^{2} \)
61 \( 1 + (-3.39 + 3.92i)T + (-8.68 - 60.3i)T^{2} \)
67 \( 1 + (12.9 - 3.81i)T + (56.3 - 36.2i)T^{2} \)
71 \( 1 + (-4.07 + 1.19i)T + (59.7 - 38.3i)T^{2} \)
73 \( 1 + (2.84 - 1.82i)T + (30.3 - 66.4i)T^{2} \)
79 \( 1 + (-0.713 - 1.56i)T + (-51.7 + 59.7i)T^{2} \)
83 \( 1 + (-0.391 + 2.71i)T + (-79.6 - 23.3i)T^{2} \)
89 \( 1 + (0.255 + 0.295i)T + (-12.6 + 88.0i)T^{2} \)
97 \( 1 + (0.166 + 1.15i)T + (-93.0 + 27.3i)T^{2} \)
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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

−11.01768548691032964803463720815, −9.968630539887345036534874633016, −9.106780976976992036191020509817, −8.621917660081518311502337655734, −7.82951218079599678645106262675, −6.11410509068722230469090689337, −5.61421624114798740050003198205, −4.42698360777891401618737170966, −3.22412021670459638768409031776, −2.01695457920474434487954783059, 0.60907735023209538803140203844, 2.77189193796347062099826415469, 3.40393016849079001527807673647, 4.76290923523639895947280647589, 6.23431019091262087880018225592, 7.12740500332335438299418603466, 7.55526598646846116494856812663, 8.575502519202398684160792856733, 10.01541581725979708924195798144, 10.49956554839908054459573543744

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