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Properties

Label 2-224-56.11-c2-0-11
Degree $2$
Conductor $224$
Sign $-0.0284 + 0.999i$
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  + (2.66 − 4.61i)3-s + (1.86 − 1.07i)5-s + (6.91 − 1.06i)7-s + (−9.71 − 16.8i)9-s + (2.62 − 4.55i)11-s + 21.4i·13-s − 11.5i·15-s + (−0.463 + 0.802i)17-s + (−2.96 − 5.13i)19-s + (13.5 − 34.7i)21-s + (−7.52 + 4.34i)23-s + (−10.1 + 17.6i)25-s − 55.6·27-s − 9.42i·29-s + (−29.8 − 17.2i)31-s + ⋯
L(s)  = 1  + (0.888 − 1.53i)3-s + (0.373 − 0.215i)5-s + (0.988 − 0.152i)7-s + (−1.07 − 1.86i)9-s + (0.239 − 0.414i)11-s + 1.64i·13-s − 0.766i·15-s + (−0.0272 + 0.0472i)17-s + (−0.156 − 0.270i)19-s + (0.644 − 1.65i)21-s + (−0.327 + 0.188i)23-s + (−0.406 + 0.704i)25-s − 2.06·27-s − 0.324i·29-s + (−0.963 − 0.556i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0284 + 0.999i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0284 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(224\)    =    \(2^{5} \cdot 7\)
Sign: $-0.0284 + 0.999i$
Analytic conductor: \(6.10355\)
Root analytic conductor: \(2.47053\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{224} (207, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 224,\ (\ :1),\ -0.0284 + 0.999i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.59848 - 1.64466i\)
\(L(\frac12)\) \(\approx\) \(1.59848 - 1.64466i\)
\(L(2)\) 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 \( 1 + (-6.91 + 1.06i)T \)
good3 \( 1 + (-2.66 + 4.61i)T + (-4.5 - 7.79i)T^{2} \)
5 \( 1 + (-1.86 + 1.07i)T + (12.5 - 21.6i)T^{2} \)
11 \( 1 + (-2.62 + 4.55i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 - 21.4iT - 169T^{2} \)
17 \( 1 + (0.463 - 0.802i)T + (-144.5 - 250. i)T^{2} \)
19 \( 1 + (2.96 + 5.13i)T + (-180.5 + 312. i)T^{2} \)
23 \( 1 + (7.52 - 4.34i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + 9.42iT - 841T^{2} \)
31 \( 1 + (29.8 + 17.2i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (11.0 - 6.40i)T + (684.5 - 1.18e3i)T^{2} \)
41 \( 1 - 43.1T + 1.68e3T^{2} \)
43 \( 1 - 41.7T + 1.84e3T^{2} \)
47 \( 1 + (39.8 - 22.9i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (-64.5 - 37.2i)T + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (-26.8 + 46.4i)T + (-1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (24.0 - 13.9i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (39.2 - 67.9i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 74.5iT - 5.04e3T^{2} \)
73 \( 1 + (16.8 - 29.1i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (-26.1 + 15.1i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 - 72.9T + 6.88e3T^{2} \)
89 \( 1 + (-27.4 - 47.4i)T + (-3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + 53.7T + 9.40e3T^{2} \)
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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

−11.85484371895990751730700320714, −11.20330668796126021811617376373, −9.380599852958993922976194152494, −8.724298112190728403613461529375, −7.72759876730605609046999434724, −6.93867776687590700309196144971, −5.81112133895433562161902907628, −4.06838734497075548737398921014, −2.27929703696035106858965262291, −1.36605083453434921383499543757, 2.32548370289706160110479165417, 3.61655452841785874109024417000, 4.76782109550315070214078409088, 5.69373357455032486373619856540, 7.67153107644832192454578473346, 8.472411464142201851907023710639, 9.398684209123105421164029574769, 10.40465382340424726584808896483, 10.78701379472959123998386417840, 12.20115487321573048964937160338

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