Properties

Label 2-552-23.13-c1-0-5
Degree $2$
Conductor $552$
Sign $0.864 - 0.501i$
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  + (0.841 + 0.540i)3-s + (−0.269 + 0.589i)5-s + (1.11 + 0.326i)7-s + (0.415 + 0.909i)9-s + (2.54 − 2.93i)11-s + (2.51 − 0.738i)13-s + (−0.544 + 0.350i)15-s + (−0.215 − 1.49i)17-s + (−0.958 + 6.66i)19-s + (0.757 + 0.874i)21-s + (2.35 + 4.17i)23-s + (2.99 + 3.46i)25-s + (−0.142 + 0.989i)27-s + (−0.224 − 1.56i)29-s + (−1.66 + 1.06i)31-s + ⋯
L(s)  = 1  + (0.485 + 0.312i)3-s + (−0.120 + 0.263i)5-s + (0.419 + 0.123i)7-s + (0.138 + 0.303i)9-s + (0.766 − 0.884i)11-s + (0.697 − 0.204i)13-s + (−0.140 + 0.0904i)15-s + (−0.0521 − 0.362i)17-s + (−0.219 + 1.52i)19-s + (0.165 + 0.190i)21-s + (0.491 + 0.870i)23-s + (0.599 + 0.692i)25-s + (−0.0273 + 0.190i)27-s + (−0.0417 − 0.290i)29-s + (−0.298 + 0.191i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.864 - 0.501i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.864 - 0.501i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(552\)    =    \(2^{3} \cdot 3 \cdot 23\)
Sign: $0.864 - 0.501i$
Analytic conductor: \(4.40774\)
Root analytic conductor: \(2.09946\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{552} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 552,\ (\ :1/2),\ 0.864 - 0.501i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.77888 + 0.478844i\)
\(L(\frac12)\) \(\approx\) \(1.77888 + 0.478844i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.841 - 0.540i)T \)
23 \( 1 + (-2.35 - 4.17i)T \)
good5 \( 1 + (0.269 - 0.589i)T + (-3.27 - 3.77i)T^{2} \)
7 \( 1 + (-1.11 - 0.326i)T + (5.88 + 3.78i)T^{2} \)
11 \( 1 + (-2.54 + 2.93i)T + (-1.56 - 10.8i)T^{2} \)
13 \( 1 + (-2.51 + 0.738i)T + (10.9 - 7.02i)T^{2} \)
17 \( 1 + (0.215 + 1.49i)T + (-16.3 + 4.78i)T^{2} \)
19 \( 1 + (0.958 - 6.66i)T + (-18.2 - 5.35i)T^{2} \)
29 \( 1 + (0.224 + 1.56i)T + (-27.8 + 8.17i)T^{2} \)
31 \( 1 + (1.66 - 1.06i)T + (12.8 - 28.1i)T^{2} \)
37 \( 1 + (1.15 + 2.53i)T + (-24.2 + 27.9i)T^{2} \)
41 \( 1 + (-3.13 + 6.85i)T + (-26.8 - 30.9i)T^{2} \)
43 \( 1 + (-6.91 - 4.44i)T + (17.8 + 39.1i)T^{2} \)
47 \( 1 + 8.08T + 47T^{2} \)
53 \( 1 + (9.09 + 2.66i)T + (44.5 + 28.6i)T^{2} \)
59 \( 1 + (-6.75 + 1.98i)T + (49.6 - 31.8i)T^{2} \)
61 \( 1 + (-2.63 + 1.69i)T + (25.3 - 55.4i)T^{2} \)
67 \( 1 + (-5.69 - 6.57i)T + (-9.53 + 66.3i)T^{2} \)
71 \( 1 + (2.29 + 2.64i)T + (-10.1 + 70.2i)T^{2} \)
73 \( 1 + (-1.63 + 11.3i)T + (-70.0 - 20.5i)T^{2} \)
79 \( 1 + (11.7 - 3.43i)T + (66.4 - 42.7i)T^{2} \)
83 \( 1 + (4.30 + 9.42i)T + (-54.3 + 62.7i)T^{2} \)
89 \( 1 + (5.15 + 3.31i)T + (36.9 + 80.9i)T^{2} \)
97 \( 1 + (-2.48 + 5.43i)T + (-63.5 - 73.3i)T^{2} \)
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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

−11.02132585665986213589136376704, −9.918144905468329085823523720322, −9.017175456197030565354678380754, −8.310943705576456290836169876182, −7.42121054173344873898280484040, −6.23721406031283926903690720127, −5.30692225309168391311125839892, −3.92294527265197899466315740197, −3.20575554706513508875173419717, −1.54056715491042115351910782917, 1.27826381266289100268666535160, 2.64114058580255824246431821898, 4.10189914547100893967721650305, 4.85557756403639152579074398407, 6.42603823908475495208973147217, 7.03707961438325964906146889449, 8.188853541380232208532662847190, 8.859342082316603873839506500283, 9.643465231134927325487006047494, 10.83656350782469142864236975415

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