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Properties

Label 2-552-24.11-c1-0-35
Degree $2$
Conductor $552$
Sign $0.991 - 0.132i$
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  + (0.242 − 1.39i)2-s + (1.04 + 1.37i)3-s + (−1.88 − 0.675i)4-s + 2.15·5-s + (2.17 − 1.12i)6-s + 2.22i·7-s + (−1.39 + 2.45i)8-s + (−0.804 + 2.89i)9-s + (0.522 − 3.00i)10-s − 0.626i·11-s + (−1.04 − 3.30i)12-s + 6.39i·13-s + (3.09 + 0.539i)14-s + (2.25 + 2.97i)15-s + (3.08 + 2.54i)16-s − 6.37i·17-s + ⋯
L(s)  = 1  + (0.171 − 0.985i)2-s + (0.604 + 0.796i)3-s + (−0.941 − 0.337i)4-s + 0.964·5-s + (0.888 − 0.459i)6-s + 0.839i·7-s + (−0.494 + 0.869i)8-s + (−0.268 + 0.963i)9-s + (0.165 − 0.949i)10-s − 0.188i·11-s + (−0.300 − 0.953i)12-s + 1.77i·13-s + (0.827 + 0.144i)14-s + (0.583 + 0.767i)15-s + (0.771 + 0.636i)16-s − 1.54i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.132i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 - 0.132i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(552\)    =    \(2^{3} \cdot 3 \cdot 23\)
Sign: $0.991 - 0.132i$
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: \((2,\ 552,\ (\ :1/2),\ 0.991 - 0.132i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.96790 + 0.130633i\)
\(L(\frac12)\) \(\approx\) \(1.96790 + 0.130633i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.242 + 1.39i)T \)
3 \( 1 + (-1.04 - 1.37i)T \)
23 \( 1 + T \)
good5 \( 1 - 2.15T + 5T^{2} \)
7 \( 1 - 2.22iT - 7T^{2} \)
11 \( 1 + 0.626iT - 11T^{2} \)
13 \( 1 - 6.39iT - 13T^{2} \)
17 \( 1 + 6.37iT - 17T^{2} \)
19 \( 1 - 5.79T + 19T^{2} \)
29 \( 1 + 2.65T + 29T^{2} \)
31 \( 1 - 8.01iT - 31T^{2} \)
37 \( 1 + 10.2iT - 37T^{2} \)
41 \( 1 + 7.00iT - 41T^{2} \)
43 \( 1 - 3.82T + 43T^{2} \)
47 \( 1 - 11.0T + 47T^{2} \)
53 \( 1 - 2.62T + 53T^{2} \)
59 \( 1 + 6.88iT - 59T^{2} \)
61 \( 1 + 11.0iT - 61T^{2} \)
67 \( 1 + 6.97T + 67T^{2} \)
71 \( 1 + 5.08T + 71T^{2} \)
73 \( 1 + 10.1T + 73T^{2} \)
79 \( 1 - 9.79iT - 79T^{2} \)
83 \( 1 + 0.202iT - 83T^{2} \)
89 \( 1 + 0.147iT - 89T^{2} \)
97 \( 1 - 5.73T + 97T^{2} \)
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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

−10.76616176644901926637759327326, −9.732949711666878477670668875960, −9.171246595448721988460861443843, −8.922559396492586305368158536534, −7.31058135727247945746345237611, −5.70110908073027771267764280439, −5.09833375774510953030609949004, −3.94239110887263962579687046956, −2.73399606541663686583041057273, −1.92559311406044312200316779662, 1.13651088140311650343658348604, 2.94960451333943213182263117460, 4.08979517525488014498729935380, 5.70003078753302480098375466322, 6.06800749829071398364014048172, 7.39083680549193173017233498386, 7.79023199774466823955820525055, 8.739987422501760829669567743140, 9.812680660432348306695188269052, 10.33961007510621697732719906710

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