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Properties

Label 2-552-23.13-c1-0-8
Degree $2$
Conductor $552$
Sign $0.547 + 0.836i$
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  + (−0.841 − 0.540i)3-s + (1.21 − 2.65i)5-s + (4.41 + 1.29i)7-s + (0.415 + 0.909i)9-s + (2.35 − 2.72i)11-s + (−4.82 + 1.41i)13-s + (−2.45 + 1.57i)15-s + (0.108 + 0.753i)17-s + (−0.368 + 2.56i)19-s + (−3.01 − 3.47i)21-s + (4.04 − 2.57i)23-s + (−2.31 − 2.66i)25-s + (0.142 − 0.989i)27-s + (0.0209 + 0.145i)29-s + (6.55 − 4.21i)31-s + ⋯
L(s)  = 1  + (−0.485 − 0.312i)3-s + (0.542 − 1.18i)5-s + (1.66 + 0.490i)7-s + (0.138 + 0.303i)9-s + (0.710 − 0.820i)11-s + (−1.33 + 0.392i)13-s + (−0.634 + 0.407i)15-s + (0.0262 + 0.182i)17-s + (−0.0844 + 0.587i)19-s + (−0.657 − 0.758i)21-s + (0.843 − 0.537i)23-s + (−0.462 − 0.533i)25-s + (0.0273 − 0.190i)27-s + (0.00389 + 0.0270i)29-s + (1.17 − 0.756i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.547 + 0.836i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.547 + 0.836i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(552\)    =    \(2^{3} \cdot 3 \cdot 23\)
Sign: $0.547 + 0.836i$
Analytic conductor: \(4.40774\)
Root analytic conductor: \(2.09946\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{552} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 552,\ (\ :1/2),\ 0.547 + 0.836i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.39882 - 0.756650i\)
\(L(\frac12)\) \(\approx\) \(1.39882 - 0.756650i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.841 + 0.540i)T \)
23 \( 1 + (-4.04 + 2.57i)T \)
good5 \( 1 + (-1.21 + 2.65i)T + (-3.27 - 3.77i)T^{2} \)
7 \( 1 + (-4.41 - 1.29i)T + (5.88 + 3.78i)T^{2} \)
11 \( 1 + (-2.35 + 2.72i)T + (-1.56 - 10.8i)T^{2} \)
13 \( 1 + (4.82 - 1.41i)T + (10.9 - 7.02i)T^{2} \)
17 \( 1 + (-0.108 - 0.753i)T + (-16.3 + 4.78i)T^{2} \)
19 \( 1 + (0.368 - 2.56i)T + (-18.2 - 5.35i)T^{2} \)
29 \( 1 + (-0.0209 - 0.145i)T + (-27.8 + 8.17i)T^{2} \)
31 \( 1 + (-6.55 + 4.21i)T + (12.8 - 28.1i)T^{2} \)
37 \( 1 + (3.02 + 6.63i)T + (-24.2 + 27.9i)T^{2} \)
41 \( 1 + (-0.0578 + 0.126i)T + (-26.8 - 30.9i)T^{2} \)
43 \( 1 + (9.96 + 6.40i)T + (17.8 + 39.1i)T^{2} \)
47 \( 1 - 2.01T + 47T^{2} \)
53 \( 1 + (1.71 + 0.502i)T + (44.5 + 28.6i)T^{2} \)
59 \( 1 + (5.62 - 1.65i)T + (49.6 - 31.8i)T^{2} \)
61 \( 1 + (4.95 - 3.18i)T + (25.3 - 55.4i)T^{2} \)
67 \( 1 + (-9.74 - 11.2i)T + (-9.53 + 66.3i)T^{2} \)
71 \( 1 + (-5.91 - 6.83i)T + (-10.1 + 70.2i)T^{2} \)
73 \( 1 + (0.115 - 0.804i)T + (-70.0 - 20.5i)T^{2} \)
79 \( 1 + (8.02 - 2.35i)T + (66.4 - 42.7i)T^{2} \)
83 \( 1 + (-3.60 - 7.88i)T + (-54.3 + 62.7i)T^{2} \)
89 \( 1 + (14.2 + 9.14i)T + (36.9 + 80.9i)T^{2} \)
97 \( 1 + (-0.896 + 1.96i)T + (-63.5 - 73.3i)T^{2} \)
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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

−10.81339562511394528948488223290, −9.666743368236664619882509466093, −8.710606149847899795016380358527, −8.215382596490582655891783473558, −7.03525405303676640108649115462, −5.75941825959522225823424956388, −5.11893902963248254743446162152, −4.35163917838570163743371726000, −2.18929994260018109364435839492, −1.14872757900150498295216483994, 1.63945713621890983732005337829, 2.99143844410983697535683234144, 4.63926908564125567631886198332, 5.05500248577448617605366468074, 6.57730397989229893261125160264, 7.15422663942297827361806455676, 8.095880092023705099232223546397, 9.479518334612365391330546347345, 10.17742814533235783977752054795, 10.89887383031979152310849127769

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