Properties

Label 2-552-184.83-c1-0-28
Degree $2$
Conductor $552$
Sign $0.999 + 0.0288i$
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  + (−1.01 + 0.987i)2-s + (0.654 + 0.755i)3-s + (0.0477 − 1.99i)4-s + (−0.131 − 0.917i)5-s + (−1.40 − 0.117i)6-s + (0.0841 − 0.184i)7-s + (1.92 + 2.07i)8-s + (−0.142 + 0.989i)9-s + (1.03 + 0.797i)10-s + (1.60 − 5.45i)11-s + (1.54 − 1.27i)12-s + (−2.77 + 1.26i)13-s + (0.0968 + 0.269i)14-s + (0.606 − 0.700i)15-s + (−3.99 − 0.191i)16-s + (2.86 − 4.46i)17-s + ⋯
L(s)  = 1  + (−0.715 + 0.698i)2-s + (0.378 + 0.436i)3-s + (0.0238 − 0.999i)4-s + (−0.0589 − 0.410i)5-s + (−0.575 − 0.0480i)6-s + (0.0317 − 0.0696i)7-s + (0.681 + 0.731i)8-s + (−0.0474 + 0.329i)9-s + (0.328 + 0.252i)10-s + (0.483 − 1.64i)11-s + (0.445 − 0.367i)12-s + (−0.769 + 0.351i)13-s + (0.0258 + 0.0720i)14-s + (0.156 − 0.180i)15-s + (−0.998 − 0.0477i)16-s + (0.695 − 1.08i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0288i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0288i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(552\)    =    \(2^{3} \cdot 3 \cdot 23\)
Sign: $0.999 + 0.0288i$
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: \((2,\ 552,\ (\ :1/2),\ 0.999 + 0.0288i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.11212 - 0.0160491i\)
\(L(\frac12)\) \(\approx\) \(1.11212 - 0.0160491i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.01 - 0.987i)T \)
3 \( 1 + (-0.654 - 0.755i)T \)
23 \( 1 + (1.13 + 4.65i)T \)
good5 \( 1 + (0.131 + 0.917i)T + (-4.79 + 1.40i)T^{2} \)
7 \( 1 + (-0.0841 + 0.184i)T + (-4.58 - 5.29i)T^{2} \)
11 \( 1 + (-1.60 + 5.45i)T + (-9.25 - 5.94i)T^{2} \)
13 \( 1 + (2.77 - 1.26i)T + (8.51 - 9.82i)T^{2} \)
17 \( 1 + (-2.86 + 4.46i)T + (-7.06 - 15.4i)T^{2} \)
19 \( 1 + (-2.36 - 3.67i)T + (-7.89 + 17.2i)T^{2} \)
29 \( 1 + (-2.14 + 3.33i)T + (-12.0 - 26.3i)T^{2} \)
31 \( 1 + (-1.22 - 1.05i)T + (4.41 + 30.6i)T^{2} \)
37 \( 1 + (-1.42 + 9.91i)T + (-35.5 - 10.4i)T^{2} \)
41 \( 1 + (-0.414 - 2.88i)T + (-39.3 + 11.5i)T^{2} \)
43 \( 1 + (1.13 - 0.985i)T + (6.11 - 42.5i)T^{2} \)
47 \( 1 - 12.0iT - 47T^{2} \)
53 \( 1 + (-5.69 + 12.4i)T + (-34.7 - 40.0i)T^{2} \)
59 \( 1 + (-2.40 - 5.26i)T + (-38.6 + 44.5i)T^{2} \)
61 \( 1 + (9.48 - 10.9i)T + (-8.68 - 60.3i)T^{2} \)
67 \( 1 + (1.82 + 6.20i)T + (-56.3 + 36.2i)T^{2} \)
71 \( 1 + (-1.62 - 5.54i)T + (-59.7 + 38.3i)T^{2} \)
73 \( 1 + (-7.53 + 4.84i)T + (30.3 - 66.4i)T^{2} \)
79 \( 1 + (-1.72 - 3.77i)T + (-51.7 + 59.7i)T^{2} \)
83 \( 1 + (6.47 + 0.930i)T + (79.6 + 23.3i)T^{2} \)
89 \( 1 + (-8.71 + 7.55i)T + (12.6 - 88.0i)T^{2} \)
97 \( 1 + (12.3 - 1.77i)T + (93.0 - 27.3i)T^{2} \)
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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

−10.53552611810997557101425739848, −9.646055478576387576101758264933, −8.980824620734484573946060117529, −8.225367059654196159767448957005, −7.41828152244999601668133274582, −6.23247111268985965097658606148, −5.35112354090952047858516751288, −4.28855403878310277028999134295, −2.78005731718570563010185735143, −0.870953259551657753020854421785, 1.47520896995109839390768777677, 2.61460794764461108155264023828, 3.70382459617519125822322918748, 4.99054869499431415119539294717, 6.76196258617745306096034565230, 7.30994143718876257071769603212, 8.143565239736129826756135658521, 9.196773437902215542790942706764, 9.911857934538688133190389258777, 10.57872133087552151810844320781

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