Properties

Label 2-552-184.83-c1-0-5
Degree $2$
Conductor $552$
Sign $0.0130 - 0.999i$
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  + (−1.32 − 0.496i)2-s + (0.654 + 0.755i)3-s + (1.50 + 1.31i)4-s + (−0.392 − 2.72i)5-s + (−0.492 − 1.32i)6-s + (−0.907 + 1.98i)7-s + (−1.34 − 2.48i)8-s + (−0.142 + 0.989i)9-s + (−0.834 + 3.80i)10-s + (−0.950 + 3.23i)11-s + (−0.00545 + 1.99i)12-s + (−4.51 + 2.06i)13-s + (2.18 − 2.18i)14-s + (1.80 − 2.08i)15-s + (0.547 + 3.96i)16-s + (−1.30 + 2.03i)17-s + ⋯
L(s)  = 1  + (−0.936 − 0.350i)2-s + (0.378 + 0.436i)3-s + (0.753 + 0.656i)4-s + (−0.175 − 1.22i)5-s + (−0.201 − 0.541i)6-s + (−0.342 + 0.750i)7-s + (−0.475 − 0.879i)8-s + (−0.0474 + 0.329i)9-s + (−0.263 + 1.20i)10-s + (−0.286 + 0.975i)11-s + (−0.00157 + 0.577i)12-s + (−1.25 + 0.572i)13-s + (0.584 − 0.582i)14-s + (0.466 − 0.538i)15-s + (0.136 + 0.990i)16-s + (−0.316 + 0.492i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0130 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0130 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(552\)    =    \(2^{3} \cdot 3 \cdot 23\)
Sign: $0.0130 - 0.999i$
Analytic conductor: \(4.40774\)
Root analytic conductor: \(2.09946\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{552} (451, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 552,\ (\ :1/2),\ 0.0130 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.480491 + 0.474243i\)
\(L(\frac12)\) \(\approx\) \(0.480491 + 0.474243i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.32 + 0.496i)T \)
3 \( 1 + (-0.654 - 0.755i)T \)
23 \( 1 + (-4.18 + 2.33i)T \)
good5 \( 1 + (0.392 + 2.72i)T + (-4.79 + 1.40i)T^{2} \)
7 \( 1 + (0.907 - 1.98i)T + (-4.58 - 5.29i)T^{2} \)
11 \( 1 + (0.950 - 3.23i)T + (-9.25 - 5.94i)T^{2} \)
13 \( 1 + (4.51 - 2.06i)T + (8.51 - 9.82i)T^{2} \)
17 \( 1 + (1.30 - 2.03i)T + (-7.06 - 15.4i)T^{2} \)
19 \( 1 + (-2.69 - 4.19i)T + (-7.89 + 17.2i)T^{2} \)
29 \( 1 + (2.47 - 3.85i)T + (-12.0 - 26.3i)T^{2} \)
31 \( 1 + (-1.92 - 1.66i)T + (4.41 + 30.6i)T^{2} \)
37 \( 1 + (-0.432 + 3.01i)T + (-35.5 - 10.4i)T^{2} \)
41 \( 1 + (-0.605 - 4.21i)T + (-39.3 + 11.5i)T^{2} \)
43 \( 1 + (7.38 - 6.39i)T + (6.11 - 42.5i)T^{2} \)
47 \( 1 - 2.58iT - 47T^{2} \)
53 \( 1 + (0.154 - 0.337i)T + (-34.7 - 40.0i)T^{2} \)
59 \( 1 + (3.80 + 8.32i)T + (-38.6 + 44.5i)T^{2} \)
61 \( 1 + (-0.0167 + 0.0193i)T + (-8.68 - 60.3i)T^{2} \)
67 \( 1 + (1.24 + 4.24i)T + (-56.3 + 36.2i)T^{2} \)
71 \( 1 + (-2.62 - 8.93i)T + (-59.7 + 38.3i)T^{2} \)
73 \( 1 + (1.85 - 1.19i)T + (30.3 - 66.4i)T^{2} \)
79 \( 1 + (-1.14 - 2.51i)T + (-51.7 + 59.7i)T^{2} \)
83 \( 1 + (-9.68 - 1.39i)T + (79.6 + 23.3i)T^{2} \)
89 \( 1 + (9.02 - 7.82i)T + (12.6 - 88.0i)T^{2} \)
97 \( 1 + (-17.0 + 2.45i)T + (93.0 - 27.3i)T^{2} \)
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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

−10.81131889470637041592210503258, −9.668440911094111420580982905339, −9.450080114870784480412912871450, −8.541804533716647068480709510428, −7.79996791925859347825364087295, −6.74455293870255228344084907631, −5.25318886929704370601293920571, −4.33837679940362204388818823843, −2.88699959638471387139670246704, −1.69865831744943190582465201467, 0.48527952086526790895734721325, 2.53937334156944037302935659250, 3.26445478103239483724997256050, 5.23425214056289811338226900974, 6.46923566233399728117350970554, 7.25875808685452202031712567015, 7.56172816387258970287645545107, 8.760715475004890576318813591564, 9.718806767510141293308397146849, 10.45157763605261662427069684716

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