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Properties

Label 2-552-23.6-c1-0-5
Degree $2$
Conductor $552$
Sign $0.817 - 0.576i$
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 + (−0.776 + 0.498i)5-s + (0.172 − 1.20i)7-s + (0.841 + 0.540i)9-s + (1.08 + 2.37i)11-s + (0.851 + 5.92i)13-s + (−0.885 + 0.259i)15-s + (4.79 − 5.53i)17-s + (3.56 + 4.11i)19-s + (0.504 − 1.10i)21-s + (1.91 − 4.39i)23-s + (−1.72 + 3.77i)25-s + (0.654 + 0.755i)27-s + (−0.971 + 1.12i)29-s + (7.26 − 2.13i)31-s + ⋯
L(s)  = 1  + (0.553 + 0.162i)3-s + (−0.347 + 0.223i)5-s + (0.0653 − 0.454i)7-s + (0.280 + 0.180i)9-s + (0.326 + 0.715i)11-s + (0.236 + 1.64i)13-s + (−0.228 + 0.0671i)15-s + (1.16 − 1.34i)17-s + (0.817 + 0.942i)19-s + (0.110 − 0.241i)21-s + (0.398 − 0.917i)23-s + (−0.344 + 0.754i)25-s + (0.126 + 0.145i)27-s + (−0.180 + 0.208i)29-s + (1.30 − 0.382i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.817 - 0.576i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.817 - 0.576i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(552\)    =    \(2^{3} \cdot 3 \cdot 23\)
Sign: $0.817 - 0.576i$
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: \((2,\ 552,\ (\ :1/2),\ 0.817 - 0.576i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.65925 + 0.526357i\)
\(L(\frac12)\) \(\approx\) \(1.65925 + 0.526357i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.91 + 4.39i)T \)
good5 \( 1 + (0.776 - 0.498i)T + (2.07 - 4.54i)T^{2} \)
7 \( 1 + (-0.172 + 1.20i)T + (-6.71 - 1.97i)T^{2} \)
11 \( 1 + (-1.08 - 2.37i)T + (-7.20 + 8.31i)T^{2} \)
13 \( 1 + (-0.851 - 5.92i)T + (-12.4 + 3.66i)T^{2} \)
17 \( 1 + (-4.79 + 5.53i)T + (-2.41 - 16.8i)T^{2} \)
19 \( 1 + (-3.56 - 4.11i)T + (-2.70 + 18.8i)T^{2} \)
29 \( 1 + (0.971 - 1.12i)T + (-4.12 - 28.7i)T^{2} \)
31 \( 1 + (-7.26 + 2.13i)T + (26.0 - 16.7i)T^{2} \)
37 \( 1 + (6.92 + 4.44i)T + (15.3 + 33.6i)T^{2} \)
41 \( 1 + (7.88 - 5.06i)T + (17.0 - 37.2i)T^{2} \)
43 \( 1 + (-7.55 - 2.21i)T + (36.1 + 23.2i)T^{2} \)
47 \( 1 + 10.1T + 47T^{2} \)
53 \( 1 + (0.545 - 3.79i)T + (-50.8 - 14.9i)T^{2} \)
59 \( 1 + (1.81 + 12.6i)T + (-56.6 + 16.6i)T^{2} \)
61 \( 1 + (5.38 - 1.57i)T + (51.3 - 32.9i)T^{2} \)
67 \( 1 + (-4.19 + 9.18i)T + (-43.8 - 50.6i)T^{2} \)
71 \( 1 + (-2.76 + 6.04i)T + (-46.4 - 53.6i)T^{2} \)
73 \( 1 + (6.43 + 7.42i)T + (-10.3 + 72.2i)T^{2} \)
79 \( 1 + (-1.74 - 12.1i)T + (-75.7 + 22.2i)T^{2} \)
83 \( 1 + (-4.39 - 2.82i)T + (34.4 + 75.4i)T^{2} \)
89 \( 1 + (3.57 + 1.05i)T + (74.8 + 48.1i)T^{2} \)
97 \( 1 + (7.58 - 4.87i)T + (40.2 - 88.2i)T^{2} \)
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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

−10.86477261283335115394548431172, −9.703816415501753170184525892724, −9.384043228532388892349614207467, −8.104324417102500868351891227141, −7.32414623104219155181627466787, −6.59536794612779561881289610990, −5.05356063614433747784725715211, −4.11064529057487763298644412389, −3.12768718603785244535776438184, −1.57839711964754017543678311302, 1.13432235227894717219688310771, 2.93670022878479880971436817419, 3.68141407014527135350879247148, 5.21594276434944386511766760552, 6.00117044336137180072512045171, 7.30269702278828505884091563891, 8.274784152679980557931953175627, 8.594009163135047221094107821100, 9.860465127078665114062626117526, 10.57086255115590524998524487954

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