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Properties

Label 2-224-8.3-c2-0-6
Degree $2$
Conductor $224$
Sign $0.728 - 0.685i$
Analytic cond. $6.10355$
Root an. cond. $2.47053$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5.22·3-s + 6.26i·5-s + 2.64i·7-s + 18.2·9-s − 9.80·11-s + 2.41i·13-s + 32.7i·15-s + 6.89·17-s − 2.77·19-s + 13.8i·21-s − 42.8i·23-s − 14.2·25-s + 48.5·27-s − 37.3i·29-s + 7.16i·31-s + ⋯
L(s)  = 1  + 1.74·3-s + 1.25i·5-s + 0.377i·7-s + 2.03·9-s − 0.891·11-s + 0.185i·13-s + 2.18i·15-s + 0.405·17-s − 0.146·19-s + 0.658i·21-s − 1.86i·23-s − 0.571·25-s + 1.79·27-s − 1.28i·29-s + 0.231i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.728 - 0.685i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.728 - 0.685i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(224\)    =    \(2^{5} \cdot 7\)
Sign: $0.728 - 0.685i$
Analytic conductor: \(6.10355\)
Root analytic conductor: \(2.47053\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{224} (15, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 224,\ (\ :1),\ 0.728 - 0.685i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.43180 + 0.964640i\)
\(L(\frac12)\) \(\approx\) \(2.43180 + 0.964640i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 - 2.64iT \)
good3 \( 1 - 5.22T + 9T^{2} \)
5 \( 1 - 6.26iT - 25T^{2} \)
11 \( 1 + 9.80T + 121T^{2} \)
13 \( 1 - 2.41iT - 169T^{2} \)
17 \( 1 - 6.89T + 289T^{2} \)
19 \( 1 + 2.77T + 361T^{2} \)
23 \( 1 + 42.8iT - 529T^{2} \)
29 \( 1 + 37.3iT - 841T^{2} \)
31 \( 1 - 7.16iT - 961T^{2} \)
37 \( 1 + 0.202iT - 1.36e3T^{2} \)
41 \( 1 - 63.5T + 1.68e3T^{2} \)
43 \( 1 - 35.3T + 1.84e3T^{2} \)
47 \( 1 - 37.9iT - 2.20e3T^{2} \)
53 \( 1 + 54.6iT - 2.80e3T^{2} \)
59 \( 1 + 104.T + 3.48e3T^{2} \)
61 \( 1 - 43.7iT - 3.72e3T^{2} \)
67 \( 1 + 31.1T + 4.48e3T^{2} \)
71 \( 1 + 23.1iT - 5.04e3T^{2} \)
73 \( 1 + 69.2T + 5.32e3T^{2} \)
79 \( 1 + 19.9iT - 6.24e3T^{2} \)
83 \( 1 - 5.11T + 6.88e3T^{2} \)
89 \( 1 + 17.9T + 7.92e3T^{2} \)
97 \( 1 - 12.4T + 9.40e3T^{2} \)
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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

−12.38371756991864968066803620581, −10.87731416287175582625852669222, −10.12680741436826501558126022288, −9.134551021381019533308484238427, −8.116575327111563883641328502253, −7.41689210770957933085629034382, −6.23508245142469231460981650455, −4.31757122362489525776129829974, −2.94545863117885836015063369937, −2.38859923264360069341101217852, 1.44845735155733057308628993800, 3.00725620442309465079541705051, 4.19803804955054356522641160768, 5.41290231415776988915866673513, 7.43162623116733445466523525915, 7.979626737224103817424772692717, 8.984468920194788893350251834922, 9.551588323858449011955957429388, 10.71125781137069463209947660053, 12.33620199682156927425527935613

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