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Properties

Label 2-552-23.18-c1-0-6
Degree $2$
Conductor $552$
Sign $0.787 + 0.616i$
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  + (0.654 − 0.755i)3-s + (−0.0891 + 0.620i)5-s + (0.0603 + 0.132i)7-s + (−0.142 − 0.989i)9-s + (1.20 − 0.355i)11-s + (2.48 − 5.43i)13-s + (0.410 + 0.473i)15-s + (1.56 − 1.00i)17-s + (1.63 + 1.04i)19-s + (0.139 + 0.0409i)21-s + (3.23 + 3.54i)23-s + (4.42 + 1.29i)25-s + (−0.841 − 0.540i)27-s + (2.33 − 1.49i)29-s + (−4.19 − 4.84i)31-s + ⋯
L(s)  = 1  + (0.378 − 0.436i)3-s + (−0.0398 + 0.277i)5-s + (0.0228 + 0.0499i)7-s + (−0.0474 − 0.329i)9-s + (0.364 − 0.107i)11-s + (0.688 − 1.50i)13-s + (0.105 + 0.122i)15-s + (0.380 − 0.244i)17-s + (0.374 + 0.240i)19-s + (0.0304 + 0.00893i)21-s + (0.674 + 0.738i)23-s + (0.884 + 0.259i)25-s + (−0.161 − 0.104i)27-s + (0.432 − 0.278i)29-s + (−0.753 − 0.870i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.787 + 0.616i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.787 + 0.616i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(552\)    =    \(2^{3} \cdot 3 \cdot 23\)
Sign: $0.787 + 0.616i$
Analytic conductor: \(4.40774\)
Root analytic conductor: \(2.09946\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{552} (409, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 552,\ (\ :1/2),\ 0.787 + 0.616i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.62218 - 0.559668i\)
\(L(\frac12)\) \(\approx\) \(1.62218 - 0.559668i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.654 + 0.755i)T \)
23 \( 1 + (-3.23 - 3.54i)T \)
good5 \( 1 + (0.0891 - 0.620i)T + (-4.79 - 1.40i)T^{2} \)
7 \( 1 + (-0.0603 - 0.132i)T + (-4.58 + 5.29i)T^{2} \)
11 \( 1 + (-1.20 + 0.355i)T + (9.25 - 5.94i)T^{2} \)
13 \( 1 + (-2.48 + 5.43i)T + (-8.51 - 9.82i)T^{2} \)
17 \( 1 + (-1.56 + 1.00i)T + (7.06 - 15.4i)T^{2} \)
19 \( 1 + (-1.63 - 1.04i)T + (7.89 + 17.2i)T^{2} \)
29 \( 1 + (-2.33 + 1.49i)T + (12.0 - 26.3i)T^{2} \)
31 \( 1 + (4.19 + 4.84i)T + (-4.41 + 30.6i)T^{2} \)
37 \( 1 + (0.658 + 4.57i)T + (-35.5 + 10.4i)T^{2} \)
41 \( 1 + (0.492 - 3.42i)T + (-39.3 - 11.5i)T^{2} \)
43 \( 1 + (2.89 - 3.33i)T + (-6.11 - 42.5i)T^{2} \)
47 \( 1 + 6.72T + 47T^{2} \)
53 \( 1 + (-3.66 - 8.03i)T + (-34.7 + 40.0i)T^{2} \)
59 \( 1 + (1.72 - 3.77i)T + (-38.6 - 44.5i)T^{2} \)
61 \( 1 + (4.82 + 5.56i)T + (-8.68 + 60.3i)T^{2} \)
67 \( 1 + (-2.36 - 0.693i)T + (56.3 + 36.2i)T^{2} \)
71 \( 1 + (9.51 + 2.79i)T + (59.7 + 38.3i)T^{2} \)
73 \( 1 + (-0.808 - 0.519i)T + (30.3 + 66.4i)T^{2} \)
79 \( 1 + (2.48 - 5.45i)T + (-51.7 - 59.7i)T^{2} \)
83 \( 1 + (-1.19 - 8.33i)T + (-79.6 + 23.3i)T^{2} \)
89 \( 1 + (6.42 - 7.41i)T + (-12.6 - 88.0i)T^{2} \)
97 \( 1 + (-0.123 + 0.857i)T + (-93.0 - 27.3i)T^{2} \)
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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

−10.74865093314756624583552136348, −9.774399011466355540637105939404, −8.854356290536044045743786812095, −7.951243565415525269705753407615, −7.23971527323741925688586020033, −6.12834109029072712100638899119, −5.24571835984965896403433691545, −3.66531749798432517332182408257, −2.83947037054302613209405180275, −1.14738606506503476949071177656, 1.55076749889998542956241920508, 3.13469033857945930185710789569, 4.23037033859200296127012142275, 5.07214069771316958206936045943, 6.43720107710198388783833073705, 7.21571381540577677042116806595, 8.658583739290287477819117968026, 8.866112170735741287811968528412, 9.960061079093151802020198140389, 10.83221612950001486696913414147

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