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Properties

Label 2-224-32.3-c2-0-8
Degree $2$
Conductor $224$
Sign $-0.911 - 0.411i$
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  + (−0.703 + 1.87i)2-s + (0.0651 − 0.157i)3-s + (−3.00 − 2.63i)4-s + (3.56 + 8.60i)5-s + (0.248 + 0.232i)6-s + (−1.87 − 1.87i)7-s + (7.05 − 3.77i)8-s + (6.34 + 6.34i)9-s + (−18.6 + 0.615i)10-s + (0.0382 + 0.0922i)11-s + (−0.610 + 0.301i)12-s + (−0.154 + 0.372i)13-s + (4.81 − 2.18i)14-s + 1.58·15-s + (2.10 + 15.8i)16-s + 8.50i·17-s + ⋯
L(s)  = 1  + (−0.351 + 0.936i)2-s + (0.0217 − 0.0524i)3-s + (−0.752 − 0.658i)4-s + (0.712 + 1.72i)5-s + (0.0414 + 0.0387i)6-s + (−0.267 − 0.267i)7-s + (0.881 − 0.472i)8-s + (0.704 + 0.704i)9-s + (−1.86 + 0.0615i)10-s + (0.00347 + 0.00838i)11-s + (−0.0509 + 0.0251i)12-s + (−0.0118 + 0.0286i)13-s + (0.344 − 0.156i)14-s + 0.105·15-s + (0.131 + 0.991i)16-s + 0.500i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.911 - 0.411i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.911 - 0.411i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(224\)    =    \(2^{5} \cdot 7\)
Sign: $-0.911 - 0.411i$
Analytic conductor: \(6.10355\)
Root analytic conductor: \(2.47053\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{224} (99, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 224,\ (\ :1),\ -0.911 - 0.411i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.252489 + 1.17381i\)
\(L(\frac12)\) \(\approx\) \(0.252489 + 1.17381i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.703 - 1.87i)T \)
7 \( 1 + (1.87 + 1.87i)T \)
good3 \( 1 + (-0.0651 + 0.157i)T + (-6.36 - 6.36i)T^{2} \)
5 \( 1 + (-3.56 - 8.60i)T + (-17.6 + 17.6i)T^{2} \)
11 \( 1 + (-0.0382 - 0.0922i)T + (-85.5 + 85.5i)T^{2} \)
13 \( 1 + (0.154 - 0.372i)T + (-119. - 119. i)T^{2} \)
17 \( 1 - 8.50iT - 289T^{2} \)
19 \( 1 + (24.2 + 10.0i)T + (255. + 255. i)T^{2} \)
23 \( 1 + (17.8 - 17.8i)T - 529iT^{2} \)
29 \( 1 + (-32.5 - 13.4i)T + (594. + 594. i)T^{2} \)
31 \( 1 - 9.33iT - 961T^{2} \)
37 \( 1 + (14.2 + 34.3i)T + (-968. + 968. i)T^{2} \)
41 \( 1 + (36.4 + 36.4i)T + 1.68e3iT^{2} \)
43 \( 1 + (-10.0 - 24.2i)T + (-1.30e3 + 1.30e3i)T^{2} \)
47 \( 1 - 45.0T + 2.20e3T^{2} \)
53 \( 1 + (-39.9 + 16.5i)T + (1.98e3 - 1.98e3i)T^{2} \)
59 \( 1 + (-49.2 + 20.3i)T + (2.46e3 - 2.46e3i)T^{2} \)
61 \( 1 + (-4.08 - 1.69i)T + (2.63e3 + 2.63e3i)T^{2} \)
67 \( 1 + (-11.5 + 27.9i)T + (-3.17e3 - 3.17e3i)T^{2} \)
71 \( 1 + (-89.5 - 89.5i)T + 5.04e3iT^{2} \)
73 \( 1 + (-102. - 102. i)T + 5.32e3iT^{2} \)
79 \( 1 - 127.T + 6.24e3T^{2} \)
83 \( 1 + (110. + 45.7i)T + (4.87e3 + 4.87e3i)T^{2} \)
89 \( 1 + (48.6 - 48.6i)T - 7.92e3iT^{2} \)
97 \( 1 - 63.3T + 9.40e3T^{2} \)
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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

−12.78588222679076779890205703517, −10.92001139786808536204897659811, −10.40080645652933749832562968397, −9.688018006305482304732645398593, −8.292570291753443538774804120148, −7.09330375782384200804369491567, −6.67520950747832295169796340573, −5.53951243222725115308320980721, −3.93708658666626318247596437952, −2.11077951240729052860178515739, 0.75728271213563461389839415366, 2.12214008032191429654553410448, 4.03825769406425329433724872807, 4.93369512921608795524949487717, 6.33630742798201530462654853617, 8.180399942165547815324873764717, 8.849120326692703636217404462424, 9.708325080463428196987465998526, 10.32545474186258276997046160685, 12.04002351454245735778935290379

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