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Properties

Label 2-552-23.12-c1-0-0
Degree $2$
Conductor $552$
Sign $-0.652 - 0.757i$
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  + (0.142 − 0.989i)3-s + (−2.56 + 0.752i)5-s + (1.17 + 1.35i)7-s + (−0.959 − 0.281i)9-s + (−2.42 − 1.55i)11-s + (−3.30 + 3.81i)13-s + (0.380 + 2.64i)15-s + (−0.712 − 1.55i)17-s + (−1.14 + 2.51i)19-s + (1.50 − 0.968i)21-s + (−3.93 + 2.74i)23-s + (1.79 − 1.15i)25-s + (−0.415 + 0.909i)27-s + (−1.36 − 2.97i)29-s + (1.48 + 10.3i)31-s + ⋯
L(s)  = 1  + (0.0821 − 0.571i)3-s + (−1.14 + 0.336i)5-s + (0.443 + 0.511i)7-s + (−0.319 − 0.0939i)9-s + (−0.730 − 0.469i)11-s + (−0.917 + 1.05i)13-s + (0.0981 + 0.682i)15-s + (−0.172 − 0.378i)17-s + (−0.263 + 0.577i)19-s + (0.328 − 0.211i)21-s + (−0.820 + 0.571i)23-s + (0.359 − 0.230i)25-s + (−0.0799 + 0.175i)27-s + (−0.252 − 0.553i)29-s + (0.267 + 1.85i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.652 - 0.757i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.652 - 0.757i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(552\)    =    \(2^{3} \cdot 3 \cdot 23\)
Sign: $-0.652 - 0.757i$
Analytic conductor: \(4.40774\)
Root analytic conductor: \(2.09946\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{552} (265, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 552,\ (\ :1/2),\ -0.652 - 0.757i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.167263 + 0.364614i\)
\(L(\frac12)\) \(\approx\) \(0.167263 + 0.364614i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.142 + 0.989i)T \)
23 \( 1 + (3.93 - 2.74i)T \)
good5 \( 1 + (2.56 - 0.752i)T + (4.20 - 2.70i)T^{2} \)
7 \( 1 + (-1.17 - 1.35i)T + (-0.996 + 6.92i)T^{2} \)
11 \( 1 + (2.42 + 1.55i)T + (4.56 + 10.0i)T^{2} \)
13 \( 1 + (3.30 - 3.81i)T + (-1.85 - 12.8i)T^{2} \)
17 \( 1 + (0.712 + 1.55i)T + (-11.1 + 12.8i)T^{2} \)
19 \( 1 + (1.14 - 2.51i)T + (-12.4 - 14.3i)T^{2} \)
29 \( 1 + (1.36 + 2.97i)T + (-18.9 + 21.9i)T^{2} \)
31 \( 1 + (-1.48 - 10.3i)T + (-29.7 + 8.73i)T^{2} \)
37 \( 1 + (5.76 + 1.69i)T + (31.1 + 20.0i)T^{2} \)
41 \( 1 + (-1.81 + 0.531i)T + (34.4 - 22.1i)T^{2} \)
43 \( 1 + (0.668 - 4.64i)T + (-41.2 - 12.1i)T^{2} \)
47 \( 1 - 2.78T + 47T^{2} \)
53 \( 1 + (-4.66 - 5.38i)T + (-7.54 + 52.4i)T^{2} \)
59 \( 1 + (4.37 - 5.04i)T + (-8.39 - 58.3i)T^{2} \)
61 \( 1 + (1.51 + 10.5i)T + (-58.5 + 17.1i)T^{2} \)
67 \( 1 + (-2.48 + 1.59i)T + (27.8 - 60.9i)T^{2} \)
71 \( 1 + (3.59 - 2.31i)T + (29.4 - 64.5i)T^{2} \)
73 \( 1 + (-3.32 + 7.27i)T + (-47.8 - 55.1i)T^{2} \)
79 \( 1 + (-7.31 + 8.44i)T + (-11.2 - 78.1i)T^{2} \)
83 \( 1 + (13.9 + 4.09i)T + (69.8 + 44.8i)T^{2} \)
89 \( 1 + (-0.245 + 1.70i)T + (-85.3 - 25.0i)T^{2} \)
97 \( 1 + (14.8 - 4.35i)T + (81.6 - 52.4i)T^{2} \)
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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

−11.32679860574557445460888417564, −10.38029055735568881897467876676, −9.158060816118725566648181763611, −8.245584294750248545202860916502, −7.59612298246618037620291066218, −6.80609902582639007386209171551, −5.57939745535865537295476259147, −4.47409074263548685902375789781, −3.24640909831664898970885882181, −2.00122610978247542050183380176, 0.21584810580627567828931380031, 2.52505250368032629227794240217, 3.93063048591031890218361496195, 4.60643497503857336741072880118, 5.55895175179446420398415095266, 7.15265715177172407759061413006, 7.898619794812160502191488956897, 8.469901979013423333472093165950, 9.760121889802797564644083218590, 10.48117168825452097002241711068

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