Properties

Label 2-552-184.85-c1-0-25
Degree $2$
Conductor $552$
Sign $0.937 + 0.347i$
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  + (−1.36 − 0.370i)2-s + (0.540 + 0.841i)3-s + (1.72 + 1.01i)4-s + (0.808 − 0.369i)5-s + (−0.426 − 1.34i)6-s + (0.759 − 0.223i)7-s + (−1.98 − 2.01i)8-s + (−0.415 + 0.909i)9-s + (−1.24 + 0.204i)10-s + (0.857 − 0.742i)11-s + (0.0824 + 1.99i)12-s + (1.80 − 6.16i)13-s + (−1.11 + 0.0230i)14-s + (0.748 + 0.480i)15-s + (1.95 + 3.48i)16-s + (0.262 − 1.82i)17-s + ⋯
L(s)  = 1  + (−0.965 − 0.261i)2-s + (0.312 + 0.485i)3-s + (0.862 + 0.505i)4-s + (0.361 − 0.165i)5-s + (−0.174 − 0.550i)6-s + (0.287 − 0.0843i)7-s + (−0.700 − 0.713i)8-s + (−0.138 + 0.303i)9-s + (−0.392 + 0.0647i)10-s + (0.258 − 0.223i)11-s + (0.0238 + 0.576i)12-s + (0.501 − 1.70i)13-s + (−0.299 + 0.00617i)14-s + (0.193 + 0.124i)15-s + (0.488 + 0.872i)16-s + (0.0637 − 0.443i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.937 + 0.347i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.937 + 0.347i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(552\)    =    \(2^{3} \cdot 3 \cdot 23\)
Sign: $0.937 + 0.347i$
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: \((2,\ 552,\ (\ :1/2),\ 0.937 + 0.347i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.17838 - 0.211607i\)
\(L(\frac12)\) \(\approx\) \(1.17838 - 0.211607i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.36 + 0.370i)T \)
3 \( 1 + (-0.540 - 0.841i)T \)
23 \( 1 + (4.76 + 0.573i)T \)
good5 \( 1 + (-0.808 + 0.369i)T + (3.27 - 3.77i)T^{2} \)
7 \( 1 + (-0.759 + 0.223i)T + (5.88 - 3.78i)T^{2} \)
11 \( 1 + (-0.857 + 0.742i)T + (1.56 - 10.8i)T^{2} \)
13 \( 1 + (-1.80 + 6.16i)T + (-10.9 - 7.02i)T^{2} \)
17 \( 1 + (-0.262 + 1.82i)T + (-16.3 - 4.78i)T^{2} \)
19 \( 1 + (-6.71 + 0.965i)T + (18.2 - 5.35i)T^{2} \)
29 \( 1 + (0.980 + 0.140i)T + (27.8 + 8.17i)T^{2} \)
31 \( 1 + (-3.39 - 2.18i)T + (12.8 + 28.1i)T^{2} \)
37 \( 1 + (-6.20 - 2.83i)T + (24.2 + 27.9i)T^{2} \)
41 \( 1 + (-0.147 - 0.323i)T + (-26.8 + 30.9i)T^{2} \)
43 \( 1 + (-2.14 - 3.33i)T + (-17.8 + 39.1i)T^{2} \)
47 \( 1 - 12.7T + 47T^{2} \)
53 \( 1 + (2.70 + 9.21i)T + (-44.5 + 28.6i)T^{2} \)
59 \( 1 + (2.36 - 8.03i)T + (-49.6 - 31.8i)T^{2} \)
61 \( 1 + (-4.32 + 6.73i)T + (-25.3 - 55.4i)T^{2} \)
67 \( 1 + (9.22 + 7.99i)T + (9.53 + 66.3i)T^{2} \)
71 \( 1 + (-1.86 + 2.15i)T + (-10.1 - 70.2i)T^{2} \)
73 \( 1 + (-0.472 - 3.28i)T + (-70.0 + 20.5i)T^{2} \)
79 \( 1 + (-6.92 - 2.03i)T + (66.4 + 42.7i)T^{2} \)
83 \( 1 + (-11.7 - 5.35i)T + (54.3 + 62.7i)T^{2} \)
89 \( 1 + (-1.13 + 0.726i)T + (36.9 - 80.9i)T^{2} \)
97 \( 1 + (3.20 + 7.02i)T + (-63.5 + 73.3i)T^{2} \)
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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

−10.55270444569758658007852099896, −9.768656445479133243086956432884, −9.176154112490963941351551780872, −8.055929661323188863412729820479, −7.65299056916871225917052563299, −6.18273442371816498879275792537, −5.24699417974531871374206603460, −3.64793682315531913900838050565, −2.71602334428493558235159157724, −1.07691540738187535447706025730, 1.41874904727271603058753014673, 2.39063084349100053035190035403, 4.05468181400157783290689109742, 5.73703647130334022918234556328, 6.45029021259420937724851262013, 7.39595599387200054073995574924, 8.132512445600670977132118461432, 9.174196504533074377658084252498, 9.636869174680799492299521556329, 10.70462472200343340966764520260

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