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Properties

Label 2-552-23.4-c1-0-2
Degree $2$
Conductor $552$
Sign $0.105 - 0.994i$
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  + (−0.959 + 0.281i)3-s + (2.00 + 1.29i)5-s + (0.340 + 2.36i)7-s + (0.841 − 0.540i)9-s + (−0.459 + 1.00i)11-s + (0.117 − 0.817i)13-s + (−2.29 − 0.672i)15-s + (−0.0178 − 0.0206i)17-s + (−2.16 + 2.49i)19-s + (−0.993 − 2.17i)21-s + (−2.91 + 3.81i)23-s + (0.291 + 0.637i)25-s + (−0.654 + 0.755i)27-s + (5.03 + 5.80i)29-s + (2.85 + 0.837i)31-s + ⋯
L(s)  = 1  + (−0.553 + 0.162i)3-s + (0.898 + 0.577i)5-s + (0.128 + 0.895i)7-s + (0.280 − 0.180i)9-s + (−0.138 + 0.303i)11-s + (0.0325 − 0.226i)13-s + (−0.591 − 0.173i)15-s + (−0.00433 − 0.00499i)17-s + (−0.496 + 0.572i)19-s + (−0.216 − 0.474i)21-s + (−0.606 + 0.794i)23-s + (0.0582 + 0.127i)25-s + (−0.126 + 0.145i)27-s + (0.934 + 1.07i)29-s + (0.512 + 0.150i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.105 - 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.105 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(552\)    =    \(2^{3} \cdot 3 \cdot 23\)
Sign: $0.105 - 0.994i$
Analytic conductor: \(4.40774\)
Root analytic conductor: \(2.09946\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{552} (73, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 552,\ (\ :1/2),\ 0.105 - 0.994i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.974324 + 0.876086i\)
\(L(\frac12)\) \(\approx\) \(0.974324 + 0.876086i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.959 - 0.281i)T \)
23 \( 1 + (2.91 - 3.81i)T \)
good5 \( 1 + (-2.00 - 1.29i)T + (2.07 + 4.54i)T^{2} \)
7 \( 1 + (-0.340 - 2.36i)T + (-6.71 + 1.97i)T^{2} \)
11 \( 1 + (0.459 - 1.00i)T + (-7.20 - 8.31i)T^{2} \)
13 \( 1 + (-0.117 + 0.817i)T + (-12.4 - 3.66i)T^{2} \)
17 \( 1 + (0.0178 + 0.0206i)T + (-2.41 + 16.8i)T^{2} \)
19 \( 1 + (2.16 - 2.49i)T + (-2.70 - 18.8i)T^{2} \)
29 \( 1 + (-5.03 - 5.80i)T + (-4.12 + 28.7i)T^{2} \)
31 \( 1 + (-2.85 - 0.837i)T + (26.0 + 16.7i)T^{2} \)
37 \( 1 + (0.444 - 0.285i)T + (15.3 - 33.6i)T^{2} \)
41 \( 1 + (-0.728 - 0.468i)T + (17.0 + 37.2i)T^{2} \)
43 \( 1 + (0.841 - 0.247i)T + (36.1 - 23.2i)T^{2} \)
47 \( 1 - 4.46T + 47T^{2} \)
53 \( 1 + (-1.44 - 10.0i)T + (-50.8 + 14.9i)T^{2} \)
59 \( 1 + (0.787 - 5.47i)T + (-56.6 - 16.6i)T^{2} \)
61 \( 1 + (3.25 + 0.956i)T + (51.3 + 32.9i)T^{2} \)
67 \( 1 + (4.92 + 10.7i)T + (-43.8 + 50.6i)T^{2} \)
71 \( 1 + (-0.565 - 1.23i)T + (-46.4 + 53.6i)T^{2} \)
73 \( 1 + (0.761 - 0.878i)T + (-10.3 - 72.2i)T^{2} \)
79 \( 1 + (-2.04 + 14.2i)T + (-75.7 - 22.2i)T^{2} \)
83 \( 1 + (3.18 - 2.04i)T + (34.4 - 75.4i)T^{2} \)
89 \( 1 + (-7.92 + 2.32i)T + (74.8 - 48.1i)T^{2} \)
97 \( 1 + (9.52 + 6.12i)T + (40.2 + 88.2i)T^{2} \)
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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

−10.77454289809378580157637495851, −10.24224519199722597146064959788, −9.393079549335392443652565228213, −8.438235407351429591814619653757, −7.25131720752728341022034183233, −6.15551657527005154483593156456, −5.68386664042138353417439119947, −4.55119134032410856888938883780, −3.00493689618842425291217315260, −1.81648675265524355108583672496, 0.835385726778916553157743010981, 2.28528907589826096488194336190, 4.09106590850896827806273829453, 4.97893068710564356666506727435, 6.02860451789956209023253064827, 6.76725160151512406427613347410, 7.909614188507808642700926572462, 8.844723428082715027336633683752, 9.901853563753543556863437416768, 10.48333453835004753499352519428

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