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Properties

Label 2-552-24.11-c1-0-45
Degree $2$
Conductor $552$
Sign $0.234 - 0.972i$
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 + (0.153 + 2.44i)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 + (−3.45 − 0.301i)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 + (0.0625 + 0.998i)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 + (−0.996 − 0.0869i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.234 - 0.972i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.234 - 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(552\)    =    \(2^{3} \cdot 3 \cdot 23\)
Sign: $0.234 - 0.972i$
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: \((2,\ 552,\ (\ :1/2),\ 0.234 - 0.972i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.69766 + 1.33692i\)
\(L(\frac12)\) \(\approx\) \(1.69766 + 1.33692i\)
\(L(\frac{3}{2})\) 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

−10.52538348943228110692150251346, −9.567898910761723003937619045670, −9.053525132917774601434704256822, −8.608132522764613083058518468272, −7.35735519099893565204031942743, −6.35126332591424631847618588740, −5.68222475542096834765940534652, −4.70694815180388765970883793216, −2.85166644368200050233033438866, −1.88455198368064639579029931597, 1.49242139755412233863612330513, 2.33456159492231074530911989678, 3.80159112694604155920535312041, 4.40749644920734246972290318507, 5.92320774850050253751646680658, 7.17081543605473732091565990999, 8.268964736241723137816935923384, 9.088665896610783871044108069409, 9.888392854018929608025390193246, 10.51411249280168477562288667736

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