Properties

Label 2-552-24.11-c1-0-86
Degree $2$
Conductor $552$
Sign $-0.999 - 0.00158i$
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  + (1.03 − 0.961i)2-s + (1.07 − 1.35i)3-s + (0.152 − 1.99i)4-s − 4.03·5-s + (−0.183 − 2.44i)6-s − 0.282i·7-s + (−1.75 − 2.21i)8-s + (−0.672 − 2.92i)9-s + (−4.18 + 3.87i)10-s + 1.95i·11-s + (−2.53 − 2.35i)12-s − 0.803i·13-s + (−0.271 − 0.293i)14-s + (−4.35 + 5.46i)15-s + (−3.95 − 0.609i)16-s − 2.01i·17-s + ⋯
L(s)  = 1  + (0.733 − 0.679i)2-s + (0.622 − 0.782i)3-s + (0.0763 − 0.997i)4-s − 1.80·5-s + (−0.0747 − 0.997i)6-s − 0.106i·7-s + (−0.621 − 0.783i)8-s + (−0.224 − 0.974i)9-s + (−1.32 + 1.22i)10-s + 0.588i·11-s + (−0.732 − 0.680i)12-s − 0.222i·13-s + (−0.0726 − 0.0784i)14-s + (−1.12 + 1.41i)15-s + (−0.988 − 0.152i)16-s − 0.489i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.00158i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.00158i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(552\)    =    \(2^{3} \cdot 3 \cdot 23\)
Sign: $-0.999 - 0.00158i$
Analytic conductor: \(4.40774\)
Root analytic conductor: \(2.09946\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{552} (323, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 552,\ (\ :1/2),\ -0.999 - 0.00158i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00122814 + 1.54513i\)
\(L(\frac12)\) \(\approx\) \(0.00122814 + 1.54513i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.03 + 0.961i)T \)
3 \( 1 + (-1.07 + 1.35i)T \)
23 \( 1 - T \)
good5 \( 1 + 4.03T + 5T^{2} \)
7 \( 1 + 0.282iT - 7T^{2} \)
11 \( 1 - 1.95iT - 11T^{2} \)
13 \( 1 + 0.803iT - 13T^{2} \)
17 \( 1 + 2.01iT - 17T^{2} \)
19 \( 1 - 1.56T + 19T^{2} \)
29 \( 1 + 3.72T + 29T^{2} \)
31 \( 1 + 3.80iT - 31T^{2} \)
37 \( 1 + 6.71iT - 37T^{2} \)
41 \( 1 + 7.30iT - 41T^{2} \)
43 \( 1 + 5.37T + 43T^{2} \)
47 \( 1 - 9.50T + 47T^{2} \)
53 \( 1 - 10.4T + 53T^{2} \)
59 \( 1 + 9.07iT - 59T^{2} \)
61 \( 1 - 10.4iT - 61T^{2} \)
67 \( 1 - 3.71T + 67T^{2} \)
71 \( 1 - 10.4T + 71T^{2} \)
73 \( 1 + 10.0T + 73T^{2} \)
79 \( 1 + 8.93iT - 79T^{2} \)
83 \( 1 - 15.8iT - 83T^{2} \)
89 \( 1 - 2.57iT - 89T^{2} \)
97 \( 1 - 1.99T + 97T^{2} \)
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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

−10.67359255210166540973403230158, −9.436506775054864569912760273123, −8.534778388122193447719994886265, −7.39514469239476604145271435545, −7.05868133170807843884426587971, −5.52620876737370874376735300949, −4.19574304727723867022338667421, −3.56622908261199778186566624177, −2.39970342539982329861493984336, −0.65658469835954825868857622263, 2.95531296728557692411770790870, 3.73035225203452756416442753350, 4.43831598184238697149742941719, 5.43652850392766551818627514342, 6.86739721789206609160270973326, 7.76388894715227191670397537236, 8.376268938218684205643955507170, 9.042798985623547303343371991868, 10.53849521888512613981996887731, 11.42310679508889097705408233048

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