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Properties

Label 2-552-24.11-c1-0-72
Degree $2$
Conductor $552$
Sign $0.613 + 0.789i$
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  + (0.210 + 1.39i)2-s + (1.69 − 0.364i)3-s + (−1.91 + 0.589i)4-s − 3.58·5-s + (0.866 + 2.29i)6-s − 4.34i·7-s + (−1.22 − 2.54i)8-s + (2.73 − 1.23i)9-s + (−0.754 − 5.00i)10-s + 0.116i·11-s + (−3.02 + 1.69i)12-s + 0.716i·13-s + (6.07 − 0.915i)14-s + (−6.06 + 1.30i)15-s + (3.30 − 2.25i)16-s − 6.18i·17-s + ⋯
L(s)  = 1  + (0.149 + 0.988i)2-s + (0.977 − 0.210i)3-s + (−0.955 + 0.294i)4-s − 1.60·5-s + (0.353 + 0.935i)6-s − 1.64i·7-s + (−0.433 − 0.900i)8-s + (0.911 − 0.411i)9-s + (−0.238 − 1.58i)10-s + 0.0350i·11-s + (−0.872 + 0.489i)12-s + 0.198i·13-s + (1.62 − 0.244i)14-s + (−1.56 + 0.337i)15-s + (0.826 − 0.563i)16-s − 1.50i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.613 + 0.789i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.613 + 0.789i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(552\)    =    \(2^{3} \cdot 3 \cdot 23\)
Sign: $0.613 + 0.789i$
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: \((2,\ 552,\ (\ :1/2),\ 0.613 + 0.789i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.981741 - 0.480208i\)
\(L(\frac12)\) \(\approx\) \(0.981741 - 0.480208i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.210 - 1.39i)T \)
3 \( 1 + (-1.69 + 0.364i)T \)
23 \( 1 + T \)
good5 \( 1 + 3.58T + 5T^{2} \)
7 \( 1 + 4.34iT - 7T^{2} \)
11 \( 1 - 0.116iT - 11T^{2} \)
13 \( 1 - 0.716iT - 13T^{2} \)
17 \( 1 + 6.18iT - 17T^{2} \)
19 \( 1 + 7.52T + 19T^{2} \)
29 \( 1 - 5.57T + 29T^{2} \)
31 \( 1 + 4.47iT - 31T^{2} \)
37 \( 1 - 0.0947iT - 37T^{2} \)
41 \( 1 - 3.09iT - 41T^{2} \)
43 \( 1 - 3.85T + 43T^{2} \)
47 \( 1 + 7.19T + 47T^{2} \)
53 \( 1 - 1.26T + 53T^{2} \)
59 \( 1 + 0.155iT - 59T^{2} \)
61 \( 1 - 0.318iT - 61T^{2} \)
67 \( 1 + 4.36T + 67T^{2} \)
71 \( 1 - 6.67T + 71T^{2} \)
73 \( 1 + 9.87T + 73T^{2} \)
79 \( 1 - 16.1iT - 79T^{2} \)
83 \( 1 + 13.0iT - 83T^{2} \)
89 \( 1 - 9.31iT - 89T^{2} \)
97 \( 1 - 6.22T + 97T^{2} \)
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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

−10.51084018819814434754542067708, −9.528320724520424629148446071223, −8.434140852726818664344729322436, −7.891229784917595243045190720040, −7.17691152784436488812603317359, −6.65741195626817062435856359722, −4.45143894654469029601450331969, −4.26106711787487805686715797360, −3.17861171835863119559835961478, −0.54455843139334901074879212875, 2.00886083620012203847235801587, 3.07923794369083969954313551559, 3.96983033472916079656110268428, 4.80685367629245072065925583922, 6.24742789222985463631579990140, 7.894293269356300946357948998849, 8.640954483845092397964650813819, 8.770329245696244791009861762874, 10.19767415756105811373967658215, 10.90608082016809015031735671346

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