Properties

Label 2-552-184.85-c1-0-16
Degree $2$
Conductor $552$
Sign $0.416 - 0.908i$
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  + (−1.30 + 0.546i)2-s + (−0.540 − 0.841i)3-s + (1.40 − 1.42i)4-s + (−0.190 + 0.0867i)5-s + (1.16 + 0.801i)6-s + (2.41 − 0.710i)7-s + (−1.04 + 2.62i)8-s + (−0.415 + 0.909i)9-s + (0.200 − 0.217i)10-s + (−4.36 + 3.78i)11-s + (−1.95 − 0.408i)12-s + (−1.69 + 5.78i)13-s + (−2.76 + 2.24i)14-s + (0.175 + 0.112i)15-s + (−0.0697 − 3.99i)16-s + (0.569 − 3.95i)17-s + ⋯
L(s)  = 1  + (−0.922 + 0.386i)2-s + (−0.312 − 0.485i)3-s + (0.700 − 0.713i)4-s + (−0.0849 + 0.0388i)5-s + (0.475 + 0.327i)6-s + (0.914 − 0.268i)7-s + (−0.370 + 0.928i)8-s + (−0.138 + 0.303i)9-s + (0.0633 − 0.0686i)10-s + (−1.31 + 1.14i)11-s + (−0.565 − 0.117i)12-s + (−0.471 + 1.60i)13-s + (−0.739 + 0.600i)14-s + (0.0453 + 0.0291i)15-s + (−0.0174 − 0.999i)16-s + (0.138 − 0.960i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.416 - 0.908i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.416 - 0.908i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(552\)    =    \(2^{3} \cdot 3 \cdot 23\)
Sign: $0.416 - 0.908i$
Analytic conductor: \(4.40774\)
Root analytic conductor: \(2.09946\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{552} (85, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 552,\ (\ :1/2),\ 0.416 - 0.908i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.643726 + 0.412974i\)
\(L(\frac12)\) \(\approx\) \(0.643726 + 0.412974i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.30 - 0.546i)T \)
3 \( 1 + (0.540 + 0.841i)T \)
23 \( 1 + (-3.93 - 2.74i)T \)
good5 \( 1 + (0.190 - 0.0867i)T + (3.27 - 3.77i)T^{2} \)
7 \( 1 + (-2.41 + 0.710i)T + (5.88 - 3.78i)T^{2} \)
11 \( 1 + (4.36 - 3.78i)T + (1.56 - 10.8i)T^{2} \)
13 \( 1 + (1.69 - 5.78i)T + (-10.9 - 7.02i)T^{2} \)
17 \( 1 + (-0.569 + 3.95i)T + (-16.3 - 4.78i)T^{2} \)
19 \( 1 + (-6.83 + 0.982i)T + (18.2 - 5.35i)T^{2} \)
29 \( 1 + (0.673 + 0.0969i)T + (27.8 + 8.17i)T^{2} \)
31 \( 1 + (-0.246 - 0.158i)T + (12.8 + 28.1i)T^{2} \)
37 \( 1 + (1.94 + 0.889i)T + (24.2 + 27.9i)T^{2} \)
41 \( 1 + (-4.71 - 10.3i)T + (-26.8 + 30.9i)T^{2} \)
43 \( 1 + (-3.10 - 4.83i)T + (-17.8 + 39.1i)T^{2} \)
47 \( 1 - 3.14T + 47T^{2} \)
53 \( 1 + (0.724 + 2.46i)T + (-44.5 + 28.6i)T^{2} \)
59 \( 1 + (2.17 - 7.41i)T + (-49.6 - 31.8i)T^{2} \)
61 \( 1 + (-6.75 + 10.5i)T + (-25.3 - 55.4i)T^{2} \)
67 \( 1 + (-2.83 - 2.45i)T + (9.53 + 66.3i)T^{2} \)
71 \( 1 + (-7.73 + 8.92i)T + (-10.1 - 70.2i)T^{2} \)
73 \( 1 + (-1.62 - 11.3i)T + (-70.0 + 20.5i)T^{2} \)
79 \( 1 + (2.27 + 0.669i)T + (66.4 + 42.7i)T^{2} \)
83 \( 1 + (-13.0 - 5.94i)T + (54.3 + 62.7i)T^{2} \)
89 \( 1 + (14.0 - 9.05i)T + (36.9 - 80.9i)T^{2} \)
97 \( 1 + (2.56 + 5.60i)T + (-63.5 + 73.3i)T^{2} \)
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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

−11.18928837282953267589482656749, −9.744229344326741870112567372224, −9.406618220577228840241037009303, −7.932201343206749130505466758450, −7.43984125795466108979838044337, −6.88145145050526578548531625835, −5.35432022570050800416883422740, −4.79664771401000934647273269521, −2.54497766363176947450800999537, −1.39296664963390161936111070109, 0.66454871265140197538370764995, 2.55419662057193482545487707396, 3.56449031507653000230689537878, 5.21127196131488331765988954681, 5.82273037730674110085992939581, 7.49803840633279330514385157291, 8.090419209453091254155215174482, 8.763509738058667772136987586378, 10.01726592016342313913321362304, 10.60202927865905802042960277000

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