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Properties

Label 2-552-24.11-c1-0-76
Degree $2$
Conductor $552$
Sign $0.979 + 0.201i$
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  + (1.39 + 0.212i)2-s + (1.67 − 0.428i)3-s + (1.90 + 0.594i)4-s − 1.58·5-s + (2.43 − 0.241i)6-s − 2.65i·7-s + (2.54 + 1.23i)8-s + (2.63 − 1.43i)9-s + (−2.20 − 0.336i)10-s + 1.71i·11-s + (3.45 + 0.180i)12-s − 4.09i·13-s + (0.565 − 3.71i)14-s + (−2.65 + 0.676i)15-s + (3.29 + 2.27i)16-s + 5.87i·17-s + ⋯
L(s)  = 1  + (0.988 + 0.150i)2-s + (0.968 − 0.247i)3-s + (0.954 + 0.297i)4-s − 0.706·5-s + (0.995 − 0.0986i)6-s − 1.00i·7-s + (0.899 + 0.437i)8-s + (0.877 − 0.479i)9-s + (−0.698 − 0.106i)10-s + 0.517i·11-s + (0.998 + 0.0520i)12-s − 1.13i·13-s + (0.151 − 0.993i)14-s + (−0.684 + 0.174i)15-s + (0.823 + 0.567i)16-s + 1.42i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.979 + 0.201i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.979 + 0.201i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(552\)    =    \(2^{3} \cdot 3 \cdot 23\)
Sign: $0.979 + 0.201i$
Analytic conductor: \(4.40774\)
Root analytic conductor: \(2.09946\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{552} (323, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 552,\ (\ :1/2),\ 0.979 + 0.201i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.26913 - 0.333089i\)
\(L(\frac12)\) \(\approx\) \(3.26913 - 0.333089i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.39 - 0.212i)T \)
3 \( 1 + (-1.67 + 0.428i)T \)
23 \( 1 - T \)
good5 \( 1 + 1.58T + 5T^{2} \)
7 \( 1 + 2.65iT - 7T^{2} \)
11 \( 1 - 1.71iT - 11T^{2} \)
13 \( 1 + 4.09iT - 13T^{2} \)
17 \( 1 - 5.87iT - 17T^{2} \)
19 \( 1 + 1.13T + 19T^{2} \)
29 \( 1 + 3.33T + 29T^{2} \)
31 \( 1 - 9.90iT - 31T^{2} \)
37 \( 1 - 4.56iT - 37T^{2} \)
41 \( 1 + 9.73iT - 41T^{2} \)
43 \( 1 + 6.27T + 43T^{2} \)
47 \( 1 + 2.62T + 47T^{2} \)
53 \( 1 + 8.73T + 53T^{2} \)
59 \( 1 - 1.29iT - 59T^{2} \)
61 \( 1 + 1.42iT - 61T^{2} \)
67 \( 1 + 8.90T + 67T^{2} \)
71 \( 1 + 5.40T + 71T^{2} \)
73 \( 1 - 8.08T + 73T^{2} \)
79 \( 1 + 4.26iT - 79T^{2} \)
83 \( 1 + 16.3iT - 83T^{2} \)
89 \( 1 - 7.01iT - 89T^{2} \)
97 \( 1 - 14.3T + 97T^{2} \)
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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

−10.62792990609957388312517165061, −10.24525027932087202556465955318, −8.633975705355549647950024789306, −7.81249994379024174295994653155, −7.29069430528592888816926557120, −6.31369658599456086965448468333, −4.85380898658635913654606019537, −3.84599520519745260157854895080, −3.27290715274387155847175215722, −1.68232464829815217057291598199, 2.06329403858931416007275281021, 3.03548238808452909887344079052, 4.05454073900169308647172868296, 4.91923422967852483913211744280, 6.13299428644209274415264315396, 7.24735890574200722594616960081, 8.045571664045295494428844082675, 9.139416570776214577642285479269, 9.794140928294641617590239873675, 11.33428518894556650414774085038

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