Properties

Degree 2
Conductor $ 2^{5} \cdot 7 $
Sign $-0.911 + 0.411i$
Motivic weight 2
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

\( d \)  =  \(2\)
\( N \)  =  \(224\)    =    \(2^{5} \cdot 7\)
\( \varepsilon \)  =  $-0.911 + 0.411i$
motivic weight  =  \(2\)
character  :  $\chi_{224} (43, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 224,\ (\ :1),\ -0.911 + 0.411i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.252489 - 1.17381i\)
\(L(\frac12)\)  \(\approx\)  \(0.252489 - 1.17381i\)
\(L(2)\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;7\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{2,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 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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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

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

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