Properties

Label 2-552-184.85-c1-0-13
Degree $2$
Conductor $552$
Sign $-0.796 - 0.605i$
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  + (1.01 + 0.980i)2-s + (−0.540 − 0.841i)3-s + (0.0768 + 1.99i)4-s + (0.756 − 0.345i)5-s + (0.274 − 1.38i)6-s + (−3.66 + 1.07i)7-s + (−1.88 + 2.11i)8-s + (−0.415 + 0.909i)9-s + (1.10 + 0.389i)10-s + (−2.56 + 2.22i)11-s + (1.63 − 1.14i)12-s + (−0.635 + 2.16i)13-s + (−4.78 − 2.49i)14-s + (−0.699 − 0.449i)15-s + (−3.98 + 0.307i)16-s + (−1.05 + 7.35i)17-s + ⋯
L(s)  = 1  + (0.720 + 0.693i)2-s + (−0.312 − 0.485i)3-s + (0.0384 + 0.999i)4-s + (0.338 − 0.154i)5-s + (0.111 − 0.566i)6-s + (−1.38 + 0.406i)7-s + (−0.665 + 0.746i)8-s + (−0.138 + 0.303i)9-s + (0.350 + 0.123i)10-s + (−0.772 + 0.669i)11-s + (0.473 − 0.330i)12-s + (−0.176 + 0.600i)13-s + (−1.27 − 0.666i)14-s + (−0.180 − 0.116i)15-s + (−0.997 + 0.0767i)16-s + (−0.256 + 1.78i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.796 - 0.605i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.796 - 0.605i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(552\)    =    \(2^{3} \cdot 3 \cdot 23\)
Sign: $-0.796 - 0.605i$
Analytic conductor: \(4.40774\)
Root analytic conductor: \(2.09946\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{552} (85, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 552,\ (\ :1/2),\ -0.796 - 0.605i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.396468 + 1.17680i\)
\(L(\frac12)\) \(\approx\) \(0.396468 + 1.17680i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.01 - 0.980i)T \)
3 \( 1 + (0.540 + 0.841i)T \)
23 \( 1 + (-2.22 + 4.25i)T \)
good5 \( 1 + (-0.756 + 0.345i)T + (3.27 - 3.77i)T^{2} \)
7 \( 1 + (3.66 - 1.07i)T + (5.88 - 3.78i)T^{2} \)
11 \( 1 + (2.56 - 2.22i)T + (1.56 - 10.8i)T^{2} \)
13 \( 1 + (0.635 - 2.16i)T + (-10.9 - 7.02i)T^{2} \)
17 \( 1 + (1.05 - 7.35i)T + (-16.3 - 4.78i)T^{2} \)
19 \( 1 + (-5.92 + 0.852i)T + (18.2 - 5.35i)T^{2} \)
29 \( 1 + (-6.91 - 0.994i)T + (27.8 + 8.17i)T^{2} \)
31 \( 1 + (5.48 + 3.52i)T + (12.8 + 28.1i)T^{2} \)
37 \( 1 + (-6.41 - 2.92i)T + (24.2 + 27.9i)T^{2} \)
41 \( 1 + (2.34 + 5.13i)T + (-26.8 + 30.9i)T^{2} \)
43 \( 1 + (-0.890 - 1.38i)T + (-17.8 + 39.1i)T^{2} \)
47 \( 1 + 6.67T + 47T^{2} \)
53 \( 1 + (2.73 + 9.32i)T + (-44.5 + 28.6i)T^{2} \)
59 \( 1 + (1.99 - 6.78i)T + (-49.6 - 31.8i)T^{2} \)
61 \( 1 + (-2.08 + 3.23i)T + (-25.3 - 55.4i)T^{2} \)
67 \( 1 + (-0.672 - 0.582i)T + (9.53 + 66.3i)T^{2} \)
71 \( 1 + (1.95 - 2.25i)T + (-10.1 - 70.2i)T^{2} \)
73 \( 1 + (-1.68 - 11.7i)T + (-70.0 + 20.5i)T^{2} \)
79 \( 1 + (-5.10 - 1.50i)T + (66.4 + 42.7i)T^{2} \)
83 \( 1 + (4.72 + 2.15i)T + (54.3 + 62.7i)T^{2} \)
89 \( 1 + (-14.8 + 9.51i)T + (36.9 - 80.9i)T^{2} \)
97 \( 1 + (-7.16 - 15.6i)T + (-63.5 + 73.3i)T^{2} \)
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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

−11.42127886316433590085388892609, −10.16653644288744076672832291883, −9.293560208934653721861149196337, −8.274592896864157798865437237863, −7.24682854684153266996613732079, −6.47073296507759902115843232695, −5.79282053996904990461488025527, −4.79880563631419395908612631596, −3.46059602157895489924344311114, −2.27043170377250431287709298891, 0.55905247212991892134263590603, 2.89808234267412880533547859352, 3.30675572063311091104360141697, 4.81127831952167644576900630764, 5.60099096063714347650824630235, 6.46259043448745058062102326076, 7.52924624499309121598852103413, 9.290845160504344369999510720921, 9.751220896055495076107263954967, 10.44047018097465870661245811112

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