Properties

Label 2-552-184.85-c1-0-34
Degree $2$
Conductor $552$
Sign $0.879 + 0.476i$
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  + (−1.02 + 0.973i)2-s + (−0.540 − 0.841i)3-s + (0.105 − 1.99i)4-s + (2.63 − 1.20i)5-s + (1.37 + 0.336i)6-s + (3.10 − 0.910i)7-s + (1.83 + 2.15i)8-s + (−0.415 + 0.909i)9-s + (−1.53 + 3.80i)10-s + (1.24 − 1.08i)11-s + (−1.73 + 0.991i)12-s + (0.884 − 3.01i)13-s + (−2.29 + 3.95i)14-s + (−2.44 − 1.56i)15-s + (−3.97 − 0.420i)16-s + (−0.685 + 4.76i)17-s + ⋯
L(s)  = 1  + (−0.725 + 0.688i)2-s + (−0.312 − 0.485i)3-s + (0.0525 − 0.998i)4-s + (1.18 − 0.539i)5-s + (0.560 + 0.137i)6-s + (1.17 − 0.344i)7-s + (0.649 + 0.760i)8-s + (−0.138 + 0.303i)9-s + (−0.485 + 1.20i)10-s + (0.376 − 0.326i)11-s + (−0.501 + 0.286i)12-s + (0.245 − 0.835i)13-s + (−0.613 + 1.05i)14-s + (−0.630 − 0.405i)15-s + (−0.994 − 0.105i)16-s + (−0.166 + 1.15i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.879 + 0.476i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.879 + 0.476i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(552\)    =    \(2^{3} \cdot 3 \cdot 23\)
Sign: $0.879 + 0.476i$
Analytic conductor: \(4.40774\)
Root analytic conductor: \(2.09946\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{552} (85, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 552,\ (\ :1/2),\ 0.879 + 0.476i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.22896 - 0.311373i\)
\(L(\frac12)\) \(\approx\) \(1.22896 - 0.311373i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.02 - 0.973i)T \)
3 \( 1 + (0.540 + 0.841i)T \)
23 \( 1 + (4.70 + 0.912i)T \)
good5 \( 1 + (-2.63 + 1.20i)T + (3.27 - 3.77i)T^{2} \)
7 \( 1 + (-3.10 + 0.910i)T + (5.88 - 3.78i)T^{2} \)
11 \( 1 + (-1.24 + 1.08i)T + (1.56 - 10.8i)T^{2} \)
13 \( 1 + (-0.884 + 3.01i)T + (-10.9 - 7.02i)T^{2} \)
17 \( 1 + (0.685 - 4.76i)T + (-16.3 - 4.78i)T^{2} \)
19 \( 1 + (-0.389 + 0.0560i)T + (18.2 - 5.35i)T^{2} \)
29 \( 1 + (-4.10 - 0.589i)T + (27.8 + 8.17i)T^{2} \)
31 \( 1 + (4.60 + 2.95i)T + (12.8 + 28.1i)T^{2} \)
37 \( 1 + (-3.39 - 1.54i)T + (24.2 + 27.9i)T^{2} \)
41 \( 1 + (-1.52 - 3.33i)T + (-26.8 + 30.9i)T^{2} \)
43 \( 1 + (5.82 + 9.06i)T + (-17.8 + 39.1i)T^{2} \)
47 \( 1 - 8.73T + 47T^{2} \)
53 \( 1 + (-3.74 - 12.7i)T + (-44.5 + 28.6i)T^{2} \)
59 \( 1 + (-1.25 + 4.26i)T + (-49.6 - 31.8i)T^{2} \)
61 \( 1 + (-3.95 + 6.15i)T + (-25.3 - 55.4i)T^{2} \)
67 \( 1 + (1.33 + 1.15i)T + (9.53 + 66.3i)T^{2} \)
71 \( 1 + (8.80 - 10.1i)T + (-10.1 - 70.2i)T^{2} \)
73 \( 1 + (-0.408 - 2.84i)T + (-70.0 + 20.5i)T^{2} \)
79 \( 1 + (1.26 + 0.370i)T + (66.4 + 42.7i)T^{2} \)
83 \( 1 + (7.35 + 3.35i)T + (54.3 + 62.7i)T^{2} \)
89 \( 1 + (0.761 - 0.489i)T + (36.9 - 80.9i)T^{2} \)
97 \( 1 + (4.43 + 9.70i)T + (-63.5 + 73.3i)T^{2} \)
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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

−10.56323297469228870857324687804, −9.832244627544609533725314049016, −8.651501319148057114153262677875, −8.204269332638147819965713271199, −7.18473736121213398884032578112, −5.95203485342915809939556798332, −5.64934052063206525186116982349, −4.43861994598532541515186646265, −2.00950772483988480876605594085, −1.11903780627735096141371121153, 1.61975891664483333095028557937, 2.56526813362670657060328087306, 4.11035832103022084146923507574, 5.15866575692299119304471272504, 6.37095294368149980250199090544, 7.34001449132905182621197685712, 8.543124174652130549007509641453, 9.339642779774175072497269858886, 9.955066566750479884013773623967, 10.78777552888125249505675197516

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