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Properties

Label 2-552-8.5-c1-0-32
Degree $2$
Conductor $552$
Sign $0.586 + 0.809i$
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  + (1.38 − 0.293i)2-s − i·3-s + (1.82 − 0.812i)4-s + 1.32i·5-s + (−0.293 − 1.38i)6-s + 0.191·7-s + (2.29 − 1.65i)8-s − 9-s + (0.388 + 1.83i)10-s − 2.99i·11-s + (−0.812 − 1.82i)12-s − 3.25i·13-s + (0.265 − 0.0563i)14-s + 1.32·15-s + (2.68 − 2.96i)16-s + 5.88·17-s + ⋯
L(s)  = 1  + (0.978 − 0.207i)2-s − 0.577i·3-s + (0.913 − 0.406i)4-s + 0.592i·5-s + (−0.119 − 0.564i)6-s + 0.0725·7-s + (0.809 − 0.586i)8-s − 0.333·9-s + (0.122 + 0.579i)10-s − 0.904i·11-s + (−0.234 − 0.527i)12-s − 0.903i·13-s + (0.0709 − 0.0150i)14-s + 0.341·15-s + (0.670 − 0.742i)16-s + 1.42·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.586 + 0.809i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.586 + 0.809i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(552\)    =    \(2^{3} \cdot 3 \cdot 23\)
Sign: $0.586 + 0.809i$
Analytic conductor: \(4.40774\)
Root analytic conductor: \(2.09946\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{552} (277, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 552,\ (\ :1/2),\ 0.586 + 0.809i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.40874 - 1.22902i\)
\(L(\frac12)\) \(\approx\) \(2.40874 - 1.22902i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.38 + 0.293i)T \)
3 \( 1 + iT \)
23 \( 1 + T \)
good5 \( 1 - 1.32iT - 5T^{2} \)
7 \( 1 - 0.191T + 7T^{2} \)
11 \( 1 + 2.99iT - 11T^{2} \)
13 \( 1 + 3.25iT - 13T^{2} \)
17 \( 1 - 5.88T + 17T^{2} \)
19 \( 1 - 2.93iT - 19T^{2} \)
29 \( 1 - 10.0iT - 29T^{2} \)
31 \( 1 + 10.0T + 31T^{2} \)
37 \( 1 - 2.32iT - 37T^{2} \)
41 \( 1 + 6.73T + 41T^{2} \)
43 \( 1 + 9.95iT - 43T^{2} \)
47 \( 1 - 3.66T + 47T^{2} \)
53 \( 1 - 8.44iT - 53T^{2} \)
59 \( 1 - 2.35iT - 59T^{2} \)
61 \( 1 - 7.37iT - 61T^{2} \)
67 \( 1 - 8.74iT - 67T^{2} \)
71 \( 1 + 1.06T + 71T^{2} \)
73 \( 1 + 15.7T + 73T^{2} \)
79 \( 1 - 2.11T + 79T^{2} \)
83 \( 1 - 5.07iT - 83T^{2} \)
89 \( 1 - 11.7T + 89T^{2} \)
97 \( 1 + 10.5T + 97T^{2} \)
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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

−10.68279194880755263400103874950, −10.30581977755473843028225330455, −8.751409810224945472155378292779, −7.64320136802238929083136801938, −6.97694726565848877518154451094, −5.81543000305981176283061753043, −5.33237308401738375486581139977, −3.56681040762728330106605123940, −2.98714204510969867671227087500, −1.39059813439874265374219043065, 1.92694772793985368636460006639, 3.40412145011214647448246367953, 4.45875359844984502782407469986, 5.08161654346262405735750957771, 6.13133725933201787914220887179, 7.22520593622493178686810946282, 8.097106236485461046036286544842, 9.272621006755584672938124394643, 10.03426802792198280691678213102, 11.16888844231260729185713879965

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