Properties

Degree 2
Conductor $ 2^{5} \cdot 7 $
Sign $-0.988 + 0.151i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.99 − 0.0176i)2-s + (−0.954 − 2.30i)3-s + (3.99 + 0.0707i)4-s + (−0.872 + 2.10i)5-s + (1.86 + 4.62i)6-s + (−1.87 + 1.87i)7-s + (−7.99 − 0.212i)8-s + (1.96 − 1.96i)9-s + (1.78 − 4.19i)10-s + (2.92 − 7.06i)11-s + (−3.65 − 9.28i)12-s + (−5.89 − 14.2i)13-s + (3.77 − 3.70i)14-s + 5.68·15-s + (15.9 + 0.566i)16-s + 17.6i·17-s + ⋯
L(s)  = 1  + (−0.999 − 0.00884i)2-s + (−0.318 − 0.768i)3-s + (0.999 + 0.0176i)4-s + (−0.174 + 0.421i)5-s + (0.311 + 0.770i)6-s + (−0.267 + 0.267i)7-s + (−0.999 − 0.0265i)8-s + (0.218 − 0.218i)9-s + (0.178 − 0.419i)10-s + (0.266 − 0.642i)11-s + (−0.304 − 0.773i)12-s + (−0.453 − 1.09i)13-s + (0.269 − 0.264i)14-s + 0.378·15-s + (0.999 + 0.0353i)16-s + 1.03i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.988 + 0.151i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.988 + 0.151i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(224\)    =    \(2^{5} \cdot 7\)
\( \varepsilon \)  =  $-0.988 + 0.151i$
motivic weight  =  \(2\)
character  :  $\chi_{224} (43, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 224,\ (\ :1),\ -0.988 + 0.151i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.0235711 - 0.309302i\)
\(L(\frac12)\)  \(\approx\)  \(0.0235711 - 0.309302i\)
\(L(2)\)   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 \( 1 + (1.99 + 0.0176i)T \)
7 \( 1 + (1.87 - 1.87i)T \)
good3 \( 1 + (0.954 + 2.30i)T + (-6.36 + 6.36i)T^{2} \)
5 \( 1 + (0.872 - 2.10i)T + (-17.6 - 17.6i)T^{2} \)
11 \( 1 + (-2.92 + 7.06i)T + (-85.5 - 85.5i)T^{2} \)
13 \( 1 + (5.89 + 14.2i)T + (-119. + 119. i)T^{2} \)
17 \( 1 - 17.6iT - 289T^{2} \)
19 \( 1 + (19.0 - 7.89i)T + (255. - 255. i)T^{2} \)
23 \( 1 + (29.6 + 29.6i)T + 529iT^{2} \)
29 \( 1 + (45.9 - 19.0i)T + (594. - 594. i)T^{2} \)
31 \( 1 + 35.6iT - 961T^{2} \)
37 \( 1 + (-17.8 + 43.2i)T + (-968. - 968. i)T^{2} \)
41 \( 1 + (54.6 - 54.6i)T - 1.68e3iT^{2} \)
43 \( 1 + (7.16 - 17.2i)T + (-1.30e3 - 1.30e3i)T^{2} \)
47 \( 1 + 26.6T + 2.20e3T^{2} \)
53 \( 1 + (30.7 + 12.7i)T + (1.98e3 + 1.98e3i)T^{2} \)
59 \( 1 + (-48.6 - 20.1i)T + (2.46e3 + 2.46e3i)T^{2} \)
61 \( 1 + (-48.9 + 20.2i)T + (2.63e3 - 2.63e3i)T^{2} \)
67 \( 1 + (49.1 + 118. i)T + (-3.17e3 + 3.17e3i)T^{2} \)
71 \( 1 + (-51.1 + 51.1i)T - 5.04e3iT^{2} \)
73 \( 1 + (39.1 - 39.1i)T - 5.32e3iT^{2} \)
79 \( 1 - 75.7T + 6.24e3T^{2} \)
83 \( 1 + (8.91 - 3.69i)T + (4.87e3 - 4.87e3i)T^{2} \)
89 \( 1 + (-19.6 - 19.6i)T + 7.92e3iT^{2} \)
97 \( 1 - 113.T + 9.40e3T^{2} \)
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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

−11.41753618972564846738419909162, −10.59541533312851309829721766767, −9.673715244479475575615023850648, −8.406335893811945464808785002055, −7.67197910369716998226381907211, −6.50195043834241991911821747193, −5.93031442007267979171637661555, −3.54417001808021134931760905812, −1.97148599322181902670372999474, −0.22870841370093461735479895435, 1.95198163361559882500883937313, 3.95738041633308642180908628533, 5.12270591776129824093653438241, 6.68180053978088951099049108199, 7.51395216164491897901567882814, 8.832824833267894181951436283288, 9.678761610247482745756117564059, 10.24492277498104975094530986563, 11.43153813511753696219678051651, 12.02050443022763566127348692270

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