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Properties

Label 2-552-23.3-c1-0-10
Degree $2$
Conductor $552$
Sign $0.00186 + 0.999i$
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  + (0.415 − 0.909i)3-s + (2.45 − 2.83i)5-s + (−2.34 + 1.50i)7-s + (−0.654 − 0.755i)9-s + (0.719 − 5.00i)11-s + (3.26 + 2.09i)13-s + (−1.55 − 3.40i)15-s + (−1.82 − 0.536i)17-s + (0.126 − 0.0372i)19-s + (0.397 + 2.76i)21-s + (−4.43 − 1.82i)23-s + (−1.28 − 8.94i)25-s + (−0.959 + 0.281i)27-s + (7.26 + 2.13i)29-s + (0.767 + 1.68i)31-s + ⋯
L(s)  = 1  + (0.239 − 0.525i)3-s + (1.09 − 1.26i)5-s + (−0.887 + 0.570i)7-s + (−0.218 − 0.251i)9-s + (0.217 − 1.50i)11-s + (0.905 + 0.582i)13-s + (−0.401 − 0.879i)15-s + (−0.443 − 0.130i)17-s + (0.0291 − 0.00855i)19-s + (0.0866 + 0.602i)21-s + (−0.925 − 0.379i)23-s + (−0.257 − 1.78i)25-s + (−0.184 + 0.0542i)27-s + (1.34 + 0.395i)29-s + (0.137 + 0.301i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.00186 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.00186 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(552\)    =    \(2^{3} \cdot 3 \cdot 23\)
Sign: $0.00186 + 0.999i$
Analytic conductor: \(4.40774\)
Root analytic conductor: \(2.09946\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{552} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 552,\ (\ :1/2),\ 0.00186 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.19028 - 1.18806i\)
\(L(\frac12)\) \(\approx\) \(1.19028 - 1.18806i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.415 + 0.909i)T \)
23 \( 1 + (4.43 + 1.82i)T \)
good5 \( 1 + (-2.45 + 2.83i)T + (-0.711 - 4.94i)T^{2} \)
7 \( 1 + (2.34 - 1.50i)T + (2.90 - 6.36i)T^{2} \)
11 \( 1 + (-0.719 + 5.00i)T + (-10.5 - 3.09i)T^{2} \)
13 \( 1 + (-3.26 - 2.09i)T + (5.40 + 11.8i)T^{2} \)
17 \( 1 + (1.82 + 0.536i)T + (14.3 + 9.19i)T^{2} \)
19 \( 1 + (-0.126 + 0.0372i)T + (15.9 - 10.2i)T^{2} \)
29 \( 1 + (-7.26 - 2.13i)T + (24.3 + 15.6i)T^{2} \)
31 \( 1 + (-0.767 - 1.68i)T + (-20.3 + 23.4i)T^{2} \)
37 \( 1 + (5.33 + 6.15i)T + (-5.26 + 36.6i)T^{2} \)
41 \( 1 + (2.28 - 2.64i)T + (-5.83 - 40.5i)T^{2} \)
43 \( 1 + (2.39 - 5.24i)T + (-28.1 - 32.4i)T^{2} \)
47 \( 1 - 1.46T + 47T^{2} \)
53 \( 1 + (-10.2 + 6.58i)T + (22.0 - 48.2i)T^{2} \)
59 \( 1 + (-10.6 - 6.82i)T + (24.5 + 53.6i)T^{2} \)
61 \( 1 + (0.878 + 1.92i)T + (-39.9 + 46.1i)T^{2} \)
67 \( 1 + (0.800 + 5.56i)T + (-64.2 + 18.8i)T^{2} \)
71 \( 1 + (0.192 + 1.34i)T + (-68.1 + 20.0i)T^{2} \)
73 \( 1 + (-8.87 + 2.60i)T + (61.4 - 39.4i)T^{2} \)
79 \( 1 + (-10.7 - 6.90i)T + (32.8 + 71.8i)T^{2} \)
83 \( 1 + (6.29 + 7.26i)T + (-11.8 + 82.1i)T^{2} \)
89 \( 1 + (6.57 - 14.3i)T + (-58.2 - 67.2i)T^{2} \)
97 \( 1 + (1.79 - 2.07i)T + (-13.8 - 96.0i)T^{2} \)
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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

−10.43541995173185204045964891771, −9.398545980316945629886078075137, −8.745598500592995870054646460873, −8.366180856628293567253488540364, −6.52709545513193894729488614098, −6.12533564670193560031174452637, −5.18345142726766666916527713710, −3.68305326585198710611911779491, −2.34041318171190284715865324205, −0.990296506524680014906654660026, 2.05454614078609301036956876898, 3.17309630867201165353782989679, 4.16425088466943056299629163965, 5.61081215590578914240521399310, 6.57995492649127512898658556017, 7.10584042535004193665588262111, 8.465430731071533440673458034010, 9.683879484189040203654338556116, 10.13671122126194258238594615235, 10.51554132450253446112540519014

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