Properties

Label 2-552-184.101-c1-0-0
Degree $2$
Conductor $552$
Sign $-0.956 - 0.292i$
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  + (−1.36 − 0.353i)2-s + (0.755 − 0.654i)3-s + (1.75 + 0.968i)4-s + (−0.708 + 0.101i)5-s + (−1.26 + 0.629i)6-s + (−1.58 + 3.47i)7-s + (−2.05 − 1.94i)8-s + (0.142 − 0.989i)9-s + (1.00 + 0.110i)10-s + (0.408 − 1.39i)11-s + (1.95 − 0.414i)12-s + (−5.37 + 2.45i)13-s + (3.40 − 4.20i)14-s + (−0.468 + 0.540i)15-s + (2.12 + 3.38i)16-s + (−2.62 − 1.68i)17-s + ⋯
L(s)  = 1  + (−0.968 − 0.249i)2-s + (0.436 − 0.378i)3-s + (0.875 + 0.484i)4-s + (−0.316 + 0.0455i)5-s + (−0.516 + 0.257i)6-s + (−0.600 + 1.31i)7-s + (−0.726 − 0.687i)8-s + (0.0474 − 0.329i)9-s + (0.318 + 0.0350i)10-s + (0.123 − 0.419i)11-s + (0.564 − 0.119i)12-s + (−1.48 + 0.680i)13-s + (0.909 − 1.12i)14-s + (−0.120 + 0.139i)15-s + (0.531 + 0.847i)16-s + (−0.637 − 0.409i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.956 - 0.292i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.956 - 0.292i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(552\)    =    \(2^{3} \cdot 3 \cdot 23\)
Sign: $-0.956 - 0.292i$
Analytic conductor: \(4.40774\)
Root analytic conductor: \(2.09946\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{552} (469, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 552,\ (\ :1/2),\ -0.956 - 0.292i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00364676 + 0.0243509i\)
\(L(\frac12)\) \(\approx\) \(0.00364676 + 0.0243509i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.36 + 0.353i)T \)
3 \( 1 + (-0.755 + 0.654i)T \)
23 \( 1 + (4.77 + 0.419i)T \)
good5 \( 1 + (0.708 - 0.101i)T + (4.79 - 1.40i)T^{2} \)
7 \( 1 + (1.58 - 3.47i)T + (-4.58 - 5.29i)T^{2} \)
11 \( 1 + (-0.408 + 1.39i)T + (-9.25 - 5.94i)T^{2} \)
13 \( 1 + (5.37 - 2.45i)T + (8.51 - 9.82i)T^{2} \)
17 \( 1 + (2.62 + 1.68i)T + (7.06 + 15.4i)T^{2} \)
19 \( 1 + (4.06 + 6.33i)T + (-7.89 + 17.2i)T^{2} \)
29 \( 1 + (-4.55 + 7.08i)T + (-12.0 - 26.3i)T^{2} \)
31 \( 1 + (-0.311 + 0.359i)T + (-4.41 - 30.6i)T^{2} \)
37 \( 1 + (-5.60 - 0.805i)T + (35.5 + 10.4i)T^{2} \)
41 \( 1 + (-1.28 - 8.93i)T + (-39.3 + 11.5i)T^{2} \)
43 \( 1 + (2.10 - 1.82i)T + (6.11 - 42.5i)T^{2} \)
47 \( 1 + 9.11T + 47T^{2} \)
53 \( 1 + (-4.16 - 1.90i)T + (34.7 + 40.0i)T^{2} \)
59 \( 1 + (1.26 - 0.578i)T + (38.6 - 44.5i)T^{2} \)
61 \( 1 + (3.93 + 3.40i)T + (8.68 + 60.3i)T^{2} \)
67 \( 1 + (-3.26 - 11.1i)T + (-56.3 + 36.2i)T^{2} \)
71 \( 1 + (13.1 - 3.87i)T + (59.7 - 38.3i)T^{2} \)
73 \( 1 + (-7.18 + 4.61i)T + (30.3 - 66.4i)T^{2} \)
79 \( 1 + (-5.49 - 12.0i)T + (-51.7 + 59.7i)T^{2} \)
83 \( 1 + (11.5 + 1.66i)T + (79.6 + 23.3i)T^{2} \)
89 \( 1 + (3.74 + 4.32i)T + (-12.6 + 88.0i)T^{2} \)
97 \( 1 + (-1.61 - 11.2i)T + (-93.0 + 27.3i)T^{2} \)
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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

−11.49319193382028007351440724989, −9.898220760757657752266818722553, −9.422292032807120593418491333735, −8.620124070749321991138449446085, −7.85513233228498768308505000923, −6.78080131652837341263639735722, −6.13412172169769914157661447048, −4.42238261931302645299592510436, −2.76864979735437263022930464962, −2.27825558788155405698931672378, 0.01646812535847290367813163265, 2.02485891875334490815527000457, 3.50198694029633392694241877302, 4.57410730552045633028233620379, 6.05932564974697712403463853934, 7.10503309139407314355607809305, 7.75028622319828131252699863628, 8.530700174630463669901042728098, 9.767806869276063624321942423804, 10.16089483177802041272042355964

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