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Properties

Label 2-552-24.11-c1-0-41
Degree $2$
Conductor $552$
Sign $-0.661 - 0.750i$
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  + (1.03 + 0.961i)2-s + (−0.303 + 1.70i)3-s + (0.150 + 1.99i)4-s + 3.63·5-s + (−1.95 + 1.47i)6-s + 2.53i·7-s + (−1.76 + 2.21i)8-s + (−2.81 − 1.03i)9-s + (3.76 + 3.49i)10-s − 0.442i·11-s + (−3.44 − 0.348i)12-s − 3.85i·13-s + (−2.43 + 2.62i)14-s + (−1.10 + 6.19i)15-s + (−3.95 + 0.600i)16-s − 3.64i·17-s + ⋯
L(s)  = 1  + (0.733 + 0.679i)2-s + (−0.175 + 0.984i)3-s + (0.0752 + 0.997i)4-s + 1.62·5-s + (−0.797 + 0.602i)6-s + 0.956i·7-s + (−0.622 + 0.782i)8-s + (−0.938 − 0.344i)9-s + (1.19 + 1.10i)10-s − 0.133i·11-s + (−0.994 − 0.100i)12-s − 1.07i·13-s + (−0.650 + 0.701i)14-s + (−0.284 + 1.60i)15-s + (−0.988 + 0.150i)16-s − 0.883i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.661 - 0.750i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.661 - 0.750i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(552\)    =    \(2^{3} \cdot 3 \cdot 23\)
Sign: $-0.661 - 0.750i$
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: \((2,\ 552,\ (\ :1/2),\ -0.661 - 0.750i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.01979 + 2.25773i\)
\(L(\frac12)\) \(\approx\) \(1.01979 + 2.25773i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.03 - 0.961i)T \)
3 \( 1 + (0.303 - 1.70i)T \)
23 \( 1 - T \)
good5 \( 1 - 3.63T + 5T^{2} \)
7 \( 1 - 2.53iT - 7T^{2} \)
11 \( 1 + 0.442iT - 11T^{2} \)
13 \( 1 + 3.85iT - 13T^{2} \)
17 \( 1 + 3.64iT - 17T^{2} \)
19 \( 1 - 6.33T + 19T^{2} \)
29 \( 1 + 10.2T + 29T^{2} \)
31 \( 1 + 3.02iT - 31T^{2} \)
37 \( 1 - 2.62iT - 37T^{2} \)
41 \( 1 + 3.97iT - 41T^{2} \)
43 \( 1 + 7.28T + 43T^{2} \)
47 \( 1 - 1.36T + 47T^{2} \)
53 \( 1 - 0.571T + 53T^{2} \)
59 \( 1 - 1.69iT - 59T^{2} \)
61 \( 1 - 6.09iT - 61T^{2} \)
67 \( 1 + 0.457T + 67T^{2} \)
71 \( 1 + 10.3T + 71T^{2} \)
73 \( 1 - 9.56T + 73T^{2} \)
79 \( 1 - 14.5iT - 79T^{2} \)
83 \( 1 - 5.33iT - 83T^{2} \)
89 \( 1 + 16.7iT - 89T^{2} \)
97 \( 1 - 13.2T + 97T^{2} \)
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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

−11.19533969328705409605860193804, −9.996330907011074565668839114241, −9.360339375643824005921890498927, −8.657315633854757666319313078834, −7.34804313531287913362355878346, −5.98980292937710311827085533630, −5.53736237470431130866499600283, −5.03178616666509025340384985008, −3.36954511623947730423062068940, −2.48971426340485044840930028078, 1.35044643028561120767253457080, 2.05634470602897784593752458124, 3.48331812749954530050525705226, 4.97204029358906339368167479779, 5.83151597008710419132036970837, 6.59427378568211252578715521505, 7.40726527009765737203220836684, 9.038042699951859316250676627592, 9.747839962012843498245439628032, 10.62761164826999482963593078111

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