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Properties

Label 2-552-23.4-c1-0-8
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: $\chi_{552} (73, \cdot )$
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

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

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