Properties

Label 2-552-23.4-c1-0-4
Degree $2$
Conductor $552$
Sign $0.631 - 0.775i$
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 + (1.92 + 1.23i)5-s + (0.378 + 2.63i)7-s + (0.841 − 0.540i)9-s + (−1.90 + 4.17i)11-s + (−0.246 + 1.71i)13-s + (2.19 + 0.643i)15-s + (−1.99 − 2.30i)17-s + (−1.69 + 1.95i)19-s + (1.10 + 2.41i)21-s + (4.51 − 1.60i)23-s + (0.0894 + 0.195i)25-s + (0.654 − 0.755i)27-s + (−5.02 − 5.79i)29-s + (1.10 + 0.323i)31-s + ⋯
L(s)  = 1  + (0.553 − 0.162i)3-s + (0.859 + 0.552i)5-s + (0.143 + 0.994i)7-s + (0.280 − 0.180i)9-s + (−0.574 + 1.25i)11-s + (−0.0682 + 0.474i)13-s + (0.565 + 0.166i)15-s + (−0.484 − 0.558i)17-s + (−0.389 + 0.449i)19-s + (0.241 + 0.527i)21-s + (0.942 − 0.334i)23-s + (0.0178 + 0.0391i)25-s + (0.126 − 0.145i)27-s + (−0.932 − 1.07i)29-s + (0.197 + 0.0580i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.631 - 0.775i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.631 - 0.775i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(552\)    =    \(2^{3} \cdot 3 \cdot 23\)
Sign: $0.631 - 0.775i$
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: \((2,\ 552,\ (\ :1/2),\ 0.631 - 0.775i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.73798 + 0.826079i\)
\(L(\frac12)\) \(\approx\) \(1.73798 + 0.826079i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-4.51 + 1.60i)T \)
good5 \( 1 + (-1.92 - 1.23i)T + (2.07 + 4.54i)T^{2} \)
7 \( 1 + (-0.378 - 2.63i)T + (-6.71 + 1.97i)T^{2} \)
11 \( 1 + (1.90 - 4.17i)T + (-7.20 - 8.31i)T^{2} \)
13 \( 1 + (0.246 - 1.71i)T + (-12.4 - 3.66i)T^{2} \)
17 \( 1 + (1.99 + 2.30i)T + (-2.41 + 16.8i)T^{2} \)
19 \( 1 + (1.69 - 1.95i)T + (-2.70 - 18.8i)T^{2} \)
29 \( 1 + (5.02 + 5.79i)T + (-4.12 + 28.7i)T^{2} \)
31 \( 1 + (-1.10 - 0.323i)T + (26.0 + 16.7i)T^{2} \)
37 \( 1 + (-7.40 + 4.75i)T + (15.3 - 33.6i)T^{2} \)
41 \( 1 + (-4.83 - 3.10i)T + (17.0 + 37.2i)T^{2} \)
43 \( 1 + (-4.05 + 1.19i)T + (36.1 - 23.2i)T^{2} \)
47 \( 1 + 2.43T + 47T^{2} \)
53 \( 1 + (1.13 + 7.87i)T + (-50.8 + 14.9i)T^{2} \)
59 \( 1 + (-0.762 + 5.30i)T + (-56.6 - 16.6i)T^{2} \)
61 \( 1 + (-10.2 - 3.01i)T + (51.3 + 32.9i)T^{2} \)
67 \( 1 + (-6.67 - 14.6i)T + (-43.8 + 50.6i)T^{2} \)
71 \( 1 + (6.49 + 14.2i)T + (-46.4 + 53.6i)T^{2} \)
73 \( 1 + (-6.10 + 7.04i)T + (-10.3 - 72.2i)T^{2} \)
79 \( 1 + (0.841 - 5.85i)T + (-75.7 - 22.2i)T^{2} \)
83 \( 1 + (3.82 - 2.45i)T + (34.4 - 75.4i)T^{2} \)
89 \( 1 + (-4.57 + 1.34i)T + (74.8 - 48.1i)T^{2} \)
97 \( 1 + (9.16 + 5.89i)T + (40.2 + 88.2i)T^{2} \)
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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

−10.83794005369998861891449656954, −9.727640437187811993461683825578, −9.389792865102314811282180911109, −8.284709875847162528325754539155, −7.29253226763382679250580605394, −6.41631468842718367424729059346, −5.41250604613407227971175318533, −4.27251798541827631343626956428, −2.53331555475120625040313139184, −2.14386311583496125656785462577, 1.12820616255812004257416198274, 2.69757019129681769291838569791, 3.88516795777022045132959308676, 5.05571677279000952708115068194, 5.95360510652327386932745402503, 7.16893884365754775887313053606, 8.129372394072635436956567223340, 8.919714517781284234679928034384, 9.694537668556607968100264032809, 10.75642830479973537954091520064

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