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Properties

Label 2-552-23.6-c1-0-4
Degree $2$
Conductor $552$
Sign $0.951 - 0.306i$
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 + (3.62 − 2.32i)5-s + (−0.646 + 4.49i)7-s + (0.841 + 0.540i)9-s + (0.125 + 0.274i)11-s + (0.987 + 6.86i)13-s + (−4.13 + 1.21i)15-s + (1.35 − 1.56i)17-s + (−0.419 − 0.484i)19-s + (1.88 − 4.13i)21-s + (3.13 − 3.62i)23-s + (5.63 − 12.3i)25-s + (−0.654 − 0.755i)27-s + (−5.09 + 5.87i)29-s + (−0.701 + 0.206i)31-s + ⋯
L(s)  = 1  + (−0.553 − 0.162i)3-s + (1.62 − 1.04i)5-s + (−0.244 + 1.70i)7-s + (0.280 + 0.180i)9-s + (0.0378 + 0.0829i)11-s + (0.273 + 1.90i)13-s + (−1.06 + 0.313i)15-s + (0.328 − 0.378i)17-s + (−0.0963 − 0.111i)19-s + (0.412 − 0.902i)21-s + (0.654 − 0.756i)23-s + (1.12 − 2.46i)25-s + (−0.126 − 0.145i)27-s + (−0.945 + 1.09i)29-s + (−0.126 + 0.0370i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.951 - 0.306i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.951 - 0.306i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(552\)    =    \(2^{3} \cdot 3 \cdot 23\)
Sign: $0.951 - 0.306i$
Analytic conductor: \(4.40774\)
Root analytic conductor: \(2.09946\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{552} (121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 552,\ (\ :1/2),\ 0.951 - 0.306i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.55995 + 0.244715i\)
\(L(\frac12)\) \(\approx\) \(1.55995 + 0.244715i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.13 + 3.62i)T \)
good5 \( 1 + (-3.62 + 2.32i)T + (2.07 - 4.54i)T^{2} \)
7 \( 1 + (0.646 - 4.49i)T + (-6.71 - 1.97i)T^{2} \)
11 \( 1 + (-0.125 - 0.274i)T + (-7.20 + 8.31i)T^{2} \)
13 \( 1 + (-0.987 - 6.86i)T + (-12.4 + 3.66i)T^{2} \)
17 \( 1 + (-1.35 + 1.56i)T + (-2.41 - 16.8i)T^{2} \)
19 \( 1 + (0.419 + 0.484i)T + (-2.70 + 18.8i)T^{2} \)
29 \( 1 + (5.09 - 5.87i)T + (-4.12 - 28.7i)T^{2} \)
31 \( 1 + (0.701 - 0.206i)T + (26.0 - 16.7i)T^{2} \)
37 \( 1 + (-1.45 - 0.933i)T + (15.3 + 33.6i)T^{2} \)
41 \( 1 + (-3.46 + 2.22i)T + (17.0 - 37.2i)T^{2} \)
43 \( 1 + (-7.95 - 2.33i)T + (36.1 + 23.2i)T^{2} \)
47 \( 1 - 4.24T + 47T^{2} \)
53 \( 1 + (-1.08 + 7.53i)T + (-50.8 - 14.9i)T^{2} \)
59 \( 1 + (0.279 + 1.94i)T + (-56.6 + 16.6i)T^{2} \)
61 \( 1 + (5.09 - 1.49i)T + (51.3 - 32.9i)T^{2} \)
67 \( 1 + (-3.79 + 8.31i)T + (-43.8 - 50.6i)T^{2} \)
71 \( 1 + (3.53 - 7.75i)T + (-46.4 - 53.6i)T^{2} \)
73 \( 1 + (0.589 + 0.679i)T + (-10.3 + 72.2i)T^{2} \)
79 \( 1 + (1.30 + 9.09i)T + (-75.7 + 22.2i)T^{2} \)
83 \( 1 + (9.04 + 5.81i)T + (34.4 + 75.4i)T^{2} \)
89 \( 1 + (-1.24 - 0.364i)T + (74.8 + 48.1i)T^{2} \)
97 \( 1 + (7.41 - 4.76i)T + (40.2 - 88.2i)T^{2} \)
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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

−10.88484471085427160462264012886, −9.568210130999027809692827197035, −9.189545040376850803247086191758, −8.632293632424328438417507774889, −6.85852141817249285841472748370, −6.04830737554551008221211253281, −5.42830094109829306532111143615, −4.58518197891278440668960625148, −2.45753060057483452036681221683, −1.58596616438331778397264690718, 1.12426613362006925401628361328, 2.86618921454993336049305910759, 3.92438151237776243959259979808, 5.53866835412178547458843016038, 5.99578450578830190652194526294, 7.07401103641315956904450727602, 7.73365900949275330670399469432, 9.454420801243221119086635322447, 10.06905555574190412141094960034, 10.73198878408683508917364242816

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