Properties

Label 2-552-24.11-c1-0-56
Degree $2$
Conductor $552$
Sign $-0.816 + 0.577i$
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  − 1.41i·2-s + (−1 − 1.41i)3-s − 2.00·4-s + 4·5-s + (−2.00 + 1.41i)6-s + 2.82i·7-s + 2.82i·8-s + (−1.00 + 2.82i)9-s − 5.65i·10-s − 5.65i·11-s + (2.00 + 2.82i)12-s − 5.65i·13-s + 4.00·14-s + (−4 − 5.65i)15-s + 4.00·16-s − 2.82i·17-s + ⋯
L(s)  = 1  − 0.999i·2-s + (−0.577 − 0.816i)3-s − 1.00·4-s + 1.78·5-s + (−0.816 + 0.577i)6-s + 1.06i·7-s + 1.00i·8-s + (−0.333 + 0.942i)9-s − 1.78i·10-s − 1.70i·11-s + (0.577 + 0.816i)12-s − 1.56i·13-s + 1.06·14-s + (−1.03 − 1.46i)15-s + 1.00·16-s − 0.685i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.816 + 0.577i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.816 + 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(552\)    =    \(2^{3} \cdot 3 \cdot 23\)
Sign: $-0.816 + 0.577i$
Analytic conductor: \(4.40774\)
Root analytic conductor: \(2.09946\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{552} (323, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 552,\ (\ :1/2),\ -0.816 + 0.577i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.411110 - 1.29346i\)
\(L(\frac12)\) \(\approx\) \(0.411110 - 1.29346i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 1.41iT \)
3 \( 1 + (1 + 1.41i)T \)
23 \( 1 + T \)
good5 \( 1 - 4T + 5T^{2} \)
7 \( 1 - 2.82iT - 7T^{2} \)
11 \( 1 + 5.65iT - 11T^{2} \)
13 \( 1 + 5.65iT - 13T^{2} \)
17 \( 1 + 2.82iT - 17T^{2} \)
19 \( 1 + 2T + 19T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 + 5.65iT - 31T^{2} \)
37 \( 1 - 2.82iT - 37T^{2} \)
41 \( 1 - 5.65iT - 41T^{2} \)
43 \( 1 + 6T + 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 - 12T + 53T^{2} \)
59 \( 1 - 2.82iT - 59T^{2} \)
61 \( 1 + 2.82iT - 61T^{2} \)
67 \( 1 - 2T + 67T^{2} \)
71 \( 1 - 8T + 71T^{2} \)
73 \( 1 + 6T + 73T^{2} \)
79 \( 1 - 2.82iT - 79T^{2} \)
83 \( 1 - 5.65iT - 83T^{2} \)
89 \( 1 - 8.48iT - 89T^{2} \)
97 \( 1 + 2T + 97T^{2} \)
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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

−10.54797172680633314239533396578, −9.755220705946178459626759727707, −8.754380700887461843055379533094, −8.122480679902390433930843126472, −6.31708903127929688712564443784, −5.61964960716596231119475507921, −5.27090920443695448049474379147, −2.96029861840373608926895350400, −2.29469906273664731539464218700, −0.899170683922299266067064534261, 1.73894201865689074860805050848, 4.03510961979211464119430070003, 4.70548370556210171918500755311, 5.64712345874069948308110651692, 6.69139225764459790538423052308, 7.00512274332638751610006117946, 8.775569190695483673964856257376, 9.479189093592101052369520249476, 10.12104354816596893106219656978, 10.57041398956212425591310601606

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