Properties

Degree 2
Conductor $ 2^{5} \cdot 7 $
Sign $-0.588 - 0.808i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.03 + 1.71i)2-s + (0.406 + 0.982i)3-s + (−1.87 − 3.53i)4-s + (0.691 − 1.66i)5-s + (−2.10 − 0.315i)6-s + (−1.87 + 1.87i)7-s + (7.98 + 0.433i)8-s + (5.56 − 5.56i)9-s + (2.14 + 2.90i)10-s + (−6.83 + 16.4i)11-s + (2.70 − 3.27i)12-s + (7.71 + 18.6i)13-s + (−1.27 − 5.13i)14-s + 1.92·15-s + (−8.97 + 13.2i)16-s + 1.25i·17-s + ⋯
L(s)  = 1  + (−0.515 + 0.856i)2-s + (0.135 + 0.327i)3-s + (−0.468 − 0.883i)4-s + (0.138 − 0.333i)5-s + (−0.350 − 0.0525i)6-s + (−0.267 + 0.267i)7-s + (0.998 + 0.0541i)8-s + (0.618 − 0.618i)9-s + (0.214 + 0.290i)10-s + (−0.621 + 1.49i)11-s + (0.225 − 0.273i)12-s + (0.593 + 1.43i)13-s + (−0.0912 − 0.366i)14-s + 0.128·15-s + (−0.561 + 0.827i)16-s + 0.0737i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.588 - 0.808i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.588 - 0.808i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(224\)    =    \(2^{5} \cdot 7\)
\( \varepsilon \)  =  $-0.588 - 0.808i$
motivic weight  =  \(2\)
character  :  $\chi_{224} (43, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 224,\ (\ :1),\ -0.588 - 0.808i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.489703 + 0.962671i\)
\(L(\frac12)\)  \(\approx\)  \(0.489703 + 0.962671i\)
\(L(2)\)   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 \( 1 + (1.03 - 1.71i)T \)
7 \( 1 + (1.87 - 1.87i)T \)
good3 \( 1 + (-0.406 - 0.982i)T + (-6.36 + 6.36i)T^{2} \)
5 \( 1 + (-0.691 + 1.66i)T + (-17.6 - 17.6i)T^{2} \)
11 \( 1 + (6.83 - 16.4i)T + (-85.5 - 85.5i)T^{2} \)
13 \( 1 + (-7.71 - 18.6i)T + (-119. + 119. i)T^{2} \)
17 \( 1 - 1.25iT - 289T^{2} \)
19 \( 1 + (8.00 - 3.31i)T + (255. - 255. i)T^{2} \)
23 \( 1 + (-4.46 - 4.46i)T + 529iT^{2} \)
29 \( 1 + (15.0 - 6.21i)T + (594. - 594. i)T^{2} \)
31 \( 1 - 1.05iT - 961T^{2} \)
37 \( 1 + (-17.3 + 41.7i)T + (-968. - 968. i)T^{2} \)
41 \( 1 + (41.0 - 41.0i)T - 1.68e3iT^{2} \)
43 \( 1 + (18.8 - 45.3i)T + (-1.30e3 - 1.30e3i)T^{2} \)
47 \( 1 + 16.4T + 2.20e3T^{2} \)
53 \( 1 + (-2.31 - 0.957i)T + (1.98e3 + 1.98e3i)T^{2} \)
59 \( 1 + (-60.1 - 24.9i)T + (2.46e3 + 2.46e3i)T^{2} \)
61 \( 1 + (-111. + 46.3i)T + (2.63e3 - 2.63e3i)T^{2} \)
67 \( 1 + (-12.6 - 30.5i)T + (-3.17e3 + 3.17e3i)T^{2} \)
71 \( 1 + (-19.6 + 19.6i)T - 5.04e3iT^{2} \)
73 \( 1 + (-60.5 + 60.5i)T - 5.32e3iT^{2} \)
79 \( 1 + 46.4T + 6.24e3T^{2} \)
83 \( 1 + (12.5 - 5.21i)T + (4.87e3 - 4.87e3i)T^{2} \)
89 \( 1 + (31.5 + 31.5i)T + 7.92e3iT^{2} \)
97 \( 1 + 36.2T + 9.40e3T^{2} \)
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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

−12.59996308959966858433294495713, −11.20192204418527485788002186568, −9.915854456695570677938594166963, −9.461357386615264949518539763943, −8.552243967664778833148099351150, −7.21776230444938831161314851829, −6.50203116297918621539619101324, −5.08022145022648682823818625074, −4.11068954034454996498900532902, −1.69693721631185057212786719608, 0.73090591788205806345096474052, 2.56173350545315549108517299495, 3.61877945320571728235022685529, 5.29825900469378695737445618581, 6.81925781261487767156917237525, 8.058645009948864775601944817038, 8.558952056627369700456327168957, 10.17259360628604194872443237886, 10.55514148766404615361860555357, 11.44412931896687883993126675651

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