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Properties

Label 2-552-24.11-c1-0-24
Degree $2$
Conductor $552$
Sign $0.878 - 0.476i$
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  + (−0.975 − 1.02i)2-s + (1.69 + 0.371i)3-s + (−0.0963 + 1.99i)4-s − 1.56·5-s + (−1.27 − 2.09i)6-s − 0.668i·7-s + (2.13 − 1.85i)8-s + (2.72 + 1.25i)9-s + (1.52 + 1.60i)10-s + 4.80i·11-s + (−0.904 + 3.34i)12-s + 4.48i·13-s + (−0.684 + 0.652i)14-s + (−2.65 − 0.581i)15-s + (−3.98 − 0.384i)16-s − 1.05i·17-s + ⋯
L(s)  = 1  + (−0.689 − 0.723i)2-s + (0.976 + 0.214i)3-s + (−0.0481 + 0.998i)4-s − 0.700·5-s + (−0.518 − 0.854i)6-s − 0.252i·7-s + (0.756 − 0.654i)8-s + (0.908 + 0.418i)9-s + (0.483 + 0.507i)10-s + 1.44i·11-s + (−0.261 + 0.965i)12-s + 1.24i·13-s + (−0.182 + 0.174i)14-s + (−0.684 − 0.150i)15-s + (−0.995 − 0.0962i)16-s − 0.255i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.878 - 0.476i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.878 - 0.476i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(552\)    =    \(2^{3} \cdot 3 \cdot 23\)
Sign: $0.878 - 0.476i$
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: \((2,\ 552,\ (\ :1/2),\ 0.878 - 0.476i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.18046 + 0.299654i\)
\(L(\frac12)\) \(\approx\) \(1.18046 + 0.299654i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.975 + 1.02i)T \)
3 \( 1 + (-1.69 - 0.371i)T \)
23 \( 1 - T \)
good5 \( 1 + 1.56T + 5T^{2} \)
7 \( 1 + 0.668iT - 7T^{2} \)
11 \( 1 - 4.80iT - 11T^{2} \)
13 \( 1 - 4.48iT - 13T^{2} \)
17 \( 1 + 1.05iT - 17T^{2} \)
19 \( 1 - 1.05T + 19T^{2} \)
29 \( 1 - 7.11T + 29T^{2} \)
31 \( 1 - 3.88iT - 31T^{2} \)
37 \( 1 + 2.44iT - 37T^{2} \)
41 \( 1 - 2.02iT - 41T^{2} \)
43 \( 1 + 3.57T + 43T^{2} \)
47 \( 1 - 7.09T + 47T^{2} \)
53 \( 1 - 0.0884T + 53T^{2} \)
59 \( 1 - 6.15iT - 59T^{2} \)
61 \( 1 - 6.69iT - 61T^{2} \)
67 \( 1 - 13.2T + 67T^{2} \)
71 \( 1 + 6.95T + 71T^{2} \)
73 \( 1 + 13.8T + 73T^{2} \)
79 \( 1 + 14.6iT - 79T^{2} \)
83 \( 1 + 15.1iT - 83T^{2} \)
89 \( 1 - 3.28iT - 89T^{2} \)
97 \( 1 + 14.8T + 97T^{2} \)
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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

−10.60641763177908122483884942575, −9.906189271377803505651464976680, −9.156970291352905662312548602449, −8.408359802169868664159855515855, −7.36568669122683598286750535471, −7.01391405370196101172513500093, −4.55924386698173131940308783386, −4.07307148642548292006983626036, −2.80669786051748160179851039532, −1.65268388943337769015435907581, 0.850991059575627083477521271906, 2.71503666341173594452571018980, 3.87871114227570641184897387859, 5.35871621284838802946535170861, 6.35286342747260205995663014479, 7.44055024200802445867896063425, 8.271598234472944972647667329205, 8.493008428450481025128577550160, 9.622147569563814288060267230060, 10.48355916539759563644872280841

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