Properties

Label 2-552-184.99-c1-0-16
Degree $2$
Conductor $552$
Sign $0.906 - 0.422i$
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  + (−0.964 + 1.03i)2-s + (−0.841 + 0.540i)3-s + (−0.139 − 1.99i)4-s + (0.533 + 1.16i)5-s + (0.252 − 1.39i)6-s + (0.243 − 0.0715i)7-s + (2.19 + 1.77i)8-s + (0.415 − 0.909i)9-s + (−1.72 − 0.574i)10-s + (1.92 − 1.67i)11-s + (1.19 + 1.60i)12-s + (1.89 − 6.45i)13-s + (−0.161 + 0.321i)14-s + (−1.07 − 0.693i)15-s + (−3.96 + 0.557i)16-s + (−4.79 − 0.689i)17-s + ⋯
L(s)  = 1  + (−0.681 + 0.731i)2-s + (−0.485 + 0.312i)3-s + (−0.0698 − 0.997i)4-s + (0.238 + 0.521i)5-s + (0.102 − 0.568i)6-s + (0.0921 − 0.0270i)7-s + (0.777 + 0.629i)8-s + (0.138 − 0.303i)9-s + (−0.544 − 0.181i)10-s + (0.581 − 0.503i)11-s + (0.345 + 0.462i)12-s + (0.525 − 1.78i)13-s + (−0.0430 + 0.0858i)14-s + (−0.278 − 0.179i)15-s + (−0.990 + 0.139i)16-s + (−1.16 − 0.167i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.906 - 0.422i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.906 - 0.422i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(552\)    =    \(2^{3} \cdot 3 \cdot 23\)
Sign: $0.906 - 0.422i$
Analytic conductor: \(4.40774\)
Root analytic conductor: \(2.09946\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{552} (283, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 552,\ (\ :1/2),\ 0.906 - 0.422i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.898912 + 0.199405i\)
\(L(\frac12)\) \(\approx\) \(0.898912 + 0.199405i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.964 - 1.03i)T \)
3 \( 1 + (0.841 - 0.540i)T \)
23 \( 1 + (-4.74 - 0.683i)T \)
good5 \( 1 + (-0.533 - 1.16i)T + (-3.27 + 3.77i)T^{2} \)
7 \( 1 + (-0.243 + 0.0715i)T + (5.88 - 3.78i)T^{2} \)
11 \( 1 + (-1.92 + 1.67i)T + (1.56 - 10.8i)T^{2} \)
13 \( 1 + (-1.89 + 6.45i)T + (-10.9 - 7.02i)T^{2} \)
17 \( 1 + (4.79 + 0.689i)T + (16.3 + 4.78i)T^{2} \)
19 \( 1 + (1.44 - 0.207i)T + (18.2 - 5.35i)T^{2} \)
29 \( 1 + (-9.93 - 1.42i)T + (27.8 + 8.17i)T^{2} \)
31 \( 1 + (2.41 - 3.75i)T + (-12.8 - 28.1i)T^{2} \)
37 \( 1 + (-3.76 + 8.24i)T + (-24.2 - 27.9i)T^{2} \)
41 \( 1 + (0.528 + 1.15i)T + (-26.8 + 30.9i)T^{2} \)
43 \( 1 + (-3.43 - 5.33i)T + (-17.8 + 39.1i)T^{2} \)
47 \( 1 - 5.36iT - 47T^{2} \)
53 \( 1 + (-7.43 + 2.18i)T + (44.5 - 28.6i)T^{2} \)
59 \( 1 + (-7.50 - 2.20i)T + (49.6 + 31.8i)T^{2} \)
61 \( 1 + (-6.43 - 4.13i)T + (25.3 + 55.4i)T^{2} \)
67 \( 1 + (11.1 + 9.68i)T + (9.53 + 66.3i)T^{2} \)
71 \( 1 + (-1.95 - 1.69i)T + (10.1 + 70.2i)T^{2} \)
73 \( 1 + (-0.424 - 2.95i)T + (-70.0 + 20.5i)T^{2} \)
79 \( 1 + (-5.82 - 1.70i)T + (66.4 + 42.7i)T^{2} \)
83 \( 1 + (12.9 + 5.92i)T + (54.3 + 62.7i)T^{2} \)
89 \( 1 + (-0.321 - 0.500i)T + (-36.9 + 80.9i)T^{2} \)
97 \( 1 + (-10.4 + 4.78i)T + (63.5 - 73.3i)T^{2} \)
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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

−10.82958107956993046286684242621, −10.03899154775366473702115270948, −8.949410229048010812253364090106, −8.315451307506047478393414585311, −7.10505293463773061658402930087, −6.34142670167910672945510315895, −5.58175478959961810287267059363, −4.49261469045696069311462667347, −2.87346285005459090101053443687, −0.870922187294241586680031605203, 1.22961490772676141558124577071, 2.28470609776383373142783358147, 4.07954830160392747115076815016, 4.80459383022420363099486896632, 6.55783091499440141762358964783, 6.96389187805078598063640747604, 8.464565340620968614706185152900, 8.963798386925175005875532583912, 9.802428646942582798828474401126, 10.86333732162193433661735957764

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