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Properties

Label 2-552-23.4-c1-0-1
Degree $2$
Conductor $552$
Sign $0.0841 - 0.996i$
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  + (0.959 − 0.281i)3-s + (−2.87 − 1.84i)5-s + (0.703 + 4.89i)7-s + (0.841 − 0.540i)9-s + (−2.09 + 4.57i)11-s + (−0.491 + 3.42i)13-s + (−3.27 − 0.962i)15-s + (−1.38 − 1.59i)17-s + (3.06 − 3.53i)19-s + (2.05 + 4.49i)21-s + (−2.82 + 3.87i)23-s + (2.76 + 6.06i)25-s + (0.654 − 0.755i)27-s + (2.07 + 2.39i)29-s + (9.36 + 2.74i)31-s + ⋯
L(s)  = 1  + (0.553 − 0.162i)3-s + (−1.28 − 0.825i)5-s + (0.265 + 1.84i)7-s + (0.280 − 0.180i)9-s + (−0.630 + 1.38i)11-s + (−0.136 + 0.948i)13-s + (−0.846 − 0.248i)15-s + (−0.335 − 0.387i)17-s + (0.703 − 0.812i)19-s + (0.448 + 0.981i)21-s + (−0.589 + 0.807i)23-s + (0.553 + 1.21i)25-s + (0.126 − 0.145i)27-s + (0.385 + 0.444i)29-s + (1.68 + 0.493i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0841 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0841 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(552\)    =    \(2^{3} \cdot 3 \cdot 23\)
Sign: $0.0841 - 0.996i$
Analytic conductor: \(4.40774\)
Root analytic conductor: \(2.09946\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{552} (73, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 552,\ (\ :1/2),\ 0.0841 - 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.822875 + 0.756333i\)
\(L(\frac12)\) \(\approx\) \(0.822875 + 0.756333i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.959 + 0.281i)T \)
23 \( 1 + (2.82 - 3.87i)T \)
good5 \( 1 + (2.87 + 1.84i)T + (2.07 + 4.54i)T^{2} \)
7 \( 1 + (-0.703 - 4.89i)T + (-6.71 + 1.97i)T^{2} \)
11 \( 1 + (2.09 - 4.57i)T + (-7.20 - 8.31i)T^{2} \)
13 \( 1 + (0.491 - 3.42i)T + (-12.4 - 3.66i)T^{2} \)
17 \( 1 + (1.38 + 1.59i)T + (-2.41 + 16.8i)T^{2} \)
19 \( 1 + (-3.06 + 3.53i)T + (-2.70 - 18.8i)T^{2} \)
29 \( 1 + (-2.07 - 2.39i)T + (-4.12 + 28.7i)T^{2} \)
31 \( 1 + (-9.36 - 2.74i)T + (26.0 + 16.7i)T^{2} \)
37 \( 1 + (2.85 - 1.83i)T + (15.3 - 33.6i)T^{2} \)
41 \( 1 + (-4.76 - 3.06i)T + (17.0 + 37.2i)T^{2} \)
43 \( 1 + (4.85 - 1.42i)T + (36.1 - 23.2i)T^{2} \)
47 \( 1 + 4.92T + 47T^{2} \)
53 \( 1 + (0.490 + 3.40i)T + (-50.8 + 14.9i)T^{2} \)
59 \( 1 + (-0.914 + 6.36i)T + (-56.6 - 16.6i)T^{2} \)
61 \( 1 + (10.2 + 3.00i)T + (51.3 + 32.9i)T^{2} \)
67 \( 1 + (4.16 + 9.12i)T + (-43.8 + 50.6i)T^{2} \)
71 \( 1 + (-4.60 - 10.0i)T + (-46.4 + 53.6i)T^{2} \)
73 \( 1 + (-0.668 + 0.771i)T + (-10.3 - 72.2i)T^{2} \)
79 \( 1 + (0.398 - 2.77i)T + (-75.7 - 22.2i)T^{2} \)
83 \( 1 + (-10.5 + 6.78i)T + (34.4 - 75.4i)T^{2} \)
89 \( 1 + (3.64 - 1.07i)T + (74.8 - 48.1i)T^{2} \)
97 \( 1 + (-8.50 - 5.46i)T + (40.2 + 88.2i)T^{2} \)
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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

−11.41103463944480695954540771336, −9.732867475454042368759086387851, −9.125457803665369504768447357815, −8.329917254686829401798579787503, −7.69089483829546684865637528464, −6.62208423867419565371612402404, −5.00205014029469294169046618185, −4.63691971540167913449850373980, −3.03856706789776735695283727803, −1.89192218121701542437846108548, 0.60822469053481844947905696595, 2.99386069289315540909259736028, 3.69301339373699371115720173230, 4.52468023657319733762444204610, 6.15774100963242492071435032836, 7.31355162622042849781451876690, 7.933832040741813044473285998733, 8.335911290350423029275118011250, 10.14115521309803735520374627433, 10.51942075449320152216167883679

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