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Properties

Label 2-552-23.12-c1-0-7
Degree $2$
Conductor $552$
Sign $0.921 + 0.387i$
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  + (−0.142 + 0.989i)3-s + (2.63 − 0.773i)5-s + (−2.50 − 2.88i)7-s + (−0.959 − 0.281i)9-s + (3.53 + 2.26i)11-s + (3.56 − 4.11i)13-s + (0.390 + 2.71i)15-s + (−2.56 − 5.61i)17-s + (−1.37 + 3.01i)19-s + (3.21 − 2.06i)21-s + (2.30 − 4.20i)23-s + (2.13 − 1.37i)25-s + (0.415 − 0.909i)27-s + (2.99 + 6.56i)29-s + (−0.166 − 1.15i)31-s + ⋯
L(s)  = 1  + (−0.0821 + 0.571i)3-s + (1.17 − 0.345i)5-s + (−0.945 − 1.09i)7-s + (−0.319 − 0.0939i)9-s + (1.06 + 0.684i)11-s + (0.990 − 1.14i)13-s + (0.100 + 0.701i)15-s + (−0.621 − 1.36i)17-s + (−0.315 + 0.691i)19-s + (0.701 − 0.450i)21-s + (0.479 − 0.877i)23-s + (0.427 − 0.274i)25-s + (0.0799 − 0.175i)27-s + (0.556 + 1.21i)29-s + (−0.0298 − 0.207i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.921 + 0.387i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.921 + 0.387i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(552\)    =    \(2^{3} \cdot 3 \cdot 23\)
Sign: $0.921 + 0.387i$
Analytic conductor: \(4.40774\)
Root analytic conductor: \(2.09946\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{552} (265, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 552,\ (\ :1/2),\ 0.921 + 0.387i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.60308 - 0.323283i\)
\(L(\frac12)\) \(\approx\) \(1.60308 - 0.323283i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.142 - 0.989i)T \)
23 \( 1 + (-2.30 + 4.20i)T \)
good5 \( 1 + (-2.63 + 0.773i)T + (4.20 - 2.70i)T^{2} \)
7 \( 1 + (2.50 + 2.88i)T + (-0.996 + 6.92i)T^{2} \)
11 \( 1 + (-3.53 - 2.26i)T + (4.56 + 10.0i)T^{2} \)
13 \( 1 + (-3.56 + 4.11i)T + (-1.85 - 12.8i)T^{2} \)
17 \( 1 + (2.56 + 5.61i)T + (-11.1 + 12.8i)T^{2} \)
19 \( 1 + (1.37 - 3.01i)T + (-12.4 - 14.3i)T^{2} \)
29 \( 1 + (-2.99 - 6.56i)T + (-18.9 + 21.9i)T^{2} \)
31 \( 1 + (0.166 + 1.15i)T + (-29.7 + 8.73i)T^{2} \)
37 \( 1 + (-10.7 - 3.16i)T + (31.1 + 20.0i)T^{2} \)
41 \( 1 + (-2.92 + 0.858i)T + (34.4 - 22.1i)T^{2} \)
43 \( 1 + (1.14 - 7.98i)T + (-41.2 - 12.1i)T^{2} \)
47 \( 1 + 6.41T + 47T^{2} \)
53 \( 1 + (5.32 + 6.14i)T + (-7.54 + 52.4i)T^{2} \)
59 \( 1 + (-1.83 + 2.12i)T + (-8.39 - 58.3i)T^{2} \)
61 \( 1 + (0.0993 + 0.690i)T + (-58.5 + 17.1i)T^{2} \)
67 \( 1 + (-5.88 + 3.78i)T + (27.8 - 60.9i)T^{2} \)
71 \( 1 + (-8.06 + 5.18i)T + (29.4 - 64.5i)T^{2} \)
73 \( 1 + (2.53 - 5.55i)T + (-47.8 - 55.1i)T^{2} \)
79 \( 1 + (10.5 - 12.1i)T + (-11.2 - 78.1i)T^{2} \)
83 \( 1 + (4.19 + 1.23i)T + (69.8 + 44.8i)T^{2} \)
89 \( 1 + (0.666 - 4.63i)T + (-85.3 - 25.0i)T^{2} \)
97 \( 1 + (10.7 - 3.14i)T + (81.6 - 52.4i)T^{2} \)
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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

−10.54075552377130133707263024526, −9.700935066993640887405115068456, −9.410214471198088151788981970566, −8.216688799218015399891545590170, −6.77827703894334518226558496706, −6.27968233966711258689159766329, −5.07247589770914620919521149466, −4.06325686215418404547758992113, −2.91508136226414420948826718898, −1.08600873272710843730542392335, 1.63299396256092928715565991903, 2.68719126746695956632838723388, 4.03038327852450224294453114863, 5.90979108072184544171139563937, 6.14644361229347110846443948036, 6.79149446337209874217973328161, 8.481311059693689630017896580314, 9.098572417287953662953843183586, 9.722880558217371202259965731294, 11.02765718255816186502691846344

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