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Properties

Label 2-552-184.83-c1-0-6
Degree $2$
Conductor $552$
Sign $-0.0177 - 0.999i$
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  + (−1.16 − 0.795i)2-s + (−0.654 − 0.755i)3-s + (0.733 + 1.86i)4-s + (0.583 + 4.05i)5-s + (0.164 + 1.40i)6-s + (−0.100 + 0.220i)7-s + (0.624 − 2.75i)8-s + (−0.142 + 0.989i)9-s + (2.54 − 5.20i)10-s + (0.534 − 1.82i)11-s + (0.926 − 1.77i)12-s + (1.81 − 0.830i)13-s + (0.292 − 0.177i)14-s + (2.68 − 3.09i)15-s + (−2.92 + 2.72i)16-s + (−3.86 + 6.01i)17-s + ⋯
L(s)  = 1  + (−0.826 − 0.562i)2-s + (−0.378 − 0.436i)3-s + (0.366 + 0.930i)4-s + (0.260 + 1.81i)5-s + (0.0669 + 0.573i)6-s + (−0.0380 + 0.0832i)7-s + (0.220 − 0.975i)8-s + (−0.0474 + 0.329i)9-s + (0.805 − 1.64i)10-s + (0.161 − 0.548i)11-s + (0.267 − 0.511i)12-s + (0.504 − 0.230i)13-s + (0.0782 − 0.0474i)14-s + (0.693 − 0.800i)15-s + (−0.731 + 0.682i)16-s + (−0.938 + 1.45i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0177 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0177 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(552\)    =    \(2^{3} \cdot 3 \cdot 23\)
Sign: $-0.0177 - 0.999i$
Analytic conductor: \(4.40774\)
Root analytic conductor: \(2.09946\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{552} (451, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 552,\ (\ :1/2),\ -0.0177 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.442949 + 0.450902i\)
\(L(\frac12)\) \(\approx\) \(0.442949 + 0.450902i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.16 + 0.795i)T \)
3 \( 1 + (0.654 + 0.755i)T \)
23 \( 1 + (1.47 - 4.56i)T \)
good5 \( 1 + (-0.583 - 4.05i)T + (-4.79 + 1.40i)T^{2} \)
7 \( 1 + (0.100 - 0.220i)T + (-4.58 - 5.29i)T^{2} \)
11 \( 1 + (-0.534 + 1.82i)T + (-9.25 - 5.94i)T^{2} \)
13 \( 1 + (-1.81 + 0.830i)T + (8.51 - 9.82i)T^{2} \)
17 \( 1 + (3.86 - 6.01i)T + (-7.06 - 15.4i)T^{2} \)
19 \( 1 + (1.69 + 2.63i)T + (-7.89 + 17.2i)T^{2} \)
29 \( 1 + (2.11 - 3.29i)T + (-12.0 - 26.3i)T^{2} \)
31 \( 1 + (-2.78 - 2.41i)T + (4.41 + 30.6i)T^{2} \)
37 \( 1 + (-1.24 + 8.66i)T + (-35.5 - 10.4i)T^{2} \)
41 \( 1 + (-0.00349 - 0.0243i)T + (-39.3 + 11.5i)T^{2} \)
43 \( 1 + (7.57 - 6.56i)T + (6.11 - 42.5i)T^{2} \)
47 \( 1 + 1.00iT - 47T^{2} \)
53 \( 1 + (5.85 - 12.8i)T + (-34.7 - 40.0i)T^{2} \)
59 \( 1 + (0.280 + 0.613i)T + (-38.6 + 44.5i)T^{2} \)
61 \( 1 + (-4.53 + 5.23i)T + (-8.68 - 60.3i)T^{2} \)
67 \( 1 + (2.78 + 9.48i)T + (-56.3 + 36.2i)T^{2} \)
71 \( 1 + (-1.31 - 4.49i)T + (-59.7 + 38.3i)T^{2} \)
73 \( 1 + (0.454 - 0.292i)T + (30.3 - 66.4i)T^{2} \)
79 \( 1 + (-5.28 - 11.5i)T + (-51.7 + 59.7i)T^{2} \)
83 \( 1 + (7.78 + 1.11i)T + (79.6 + 23.3i)T^{2} \)
89 \( 1 + (1.79 - 1.55i)T + (12.6 - 88.0i)T^{2} \)
97 \( 1 + (5.35 - 0.769i)T + (93.0 - 27.3i)T^{2} \)
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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

−10.97102853621547731940365097297, −10.46736193588570930964198360931, −9.386048755713897327591990116654, −8.339389663266251211936283487409, −7.41815511599716424713480915337, −6.56266831367342900191598373577, −6.00595706353548935256949675332, −3.87671626425625425963794232828, −2.90419336068889935063369170255, −1.78438626229972592096051724502, 0.49049896714482073915085572929, 1.92937900426665198739774284174, 4.37329367040367725230387306758, 4.96939665771609350186919825788, 5.96605708968754172684830280732, 6.91951440925467382003848365092, 8.267138842791763070285433003262, 8.750261682123392364766881696012, 9.633757166832116637316394076857, 10.12007305199477909709744613984

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