Properties

Degree 2
Conductor $ 2^{5} \cdot 7 $
Sign $-0.864 + 0.502i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.491 + 1.93i)2-s + (−1.80 − 4.35i)3-s + (−3.51 − 1.90i)4-s + (3.60 − 8.70i)5-s + (9.32 − 1.35i)6-s + (−1.87 + 1.87i)7-s + (5.42 − 5.88i)8-s + (−9.34 + 9.34i)9-s + (15.1 + 11.2i)10-s + (−2.46 + 5.94i)11-s + (−1.95 + 18.7i)12-s + (−7.20 − 17.3i)13-s + (−2.70 − 4.54i)14-s − 44.4·15-s + (8.74 + 13.4i)16-s + 23.3i·17-s + ⋯
L(s)  = 1  + (−0.245 + 0.969i)2-s + (−0.601 − 1.45i)3-s + (−0.879 − 0.476i)4-s + (0.721 − 1.74i)5-s + (1.55 − 0.226i)6-s + (−0.267 + 0.267i)7-s + (0.677 − 0.735i)8-s + (−1.03 + 1.03i)9-s + (1.51 + 1.12i)10-s + (−0.223 + 0.540i)11-s + (−0.162 + 1.56i)12-s + (−0.553 − 1.33i)13-s + (−0.193 − 0.324i)14-s − 2.96·15-s + (0.546 + 0.837i)16-s + 1.37i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.864 + 0.502i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.864 + 0.502i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(224\)    =    \(2^{5} \cdot 7\)
\( \varepsilon \)  =  $-0.864 + 0.502i$
motivic weight  =  \(2\)
character  :  $\chi_{224} (43, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 224,\ (\ :1),\ -0.864 + 0.502i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.185043 - 0.687072i\)
\(L(\frac12)\)  \(\approx\)  \(0.185043 - 0.687072i\)
\(L(2)\)   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 \( 1 + (0.491 - 1.93i)T \)
7 \( 1 + (1.87 - 1.87i)T \)
good3 \( 1 + (1.80 + 4.35i)T + (-6.36 + 6.36i)T^{2} \)
5 \( 1 + (-3.60 + 8.70i)T + (-17.6 - 17.6i)T^{2} \)
11 \( 1 + (2.46 - 5.94i)T + (-85.5 - 85.5i)T^{2} \)
13 \( 1 + (7.20 + 17.3i)T + (-119. + 119. i)T^{2} \)
17 \( 1 - 23.3iT - 289T^{2} \)
19 \( 1 + (-4.65 + 1.92i)T + (255. - 255. i)T^{2} \)
23 \( 1 + (8.77 + 8.77i)T + 529iT^{2} \)
29 \( 1 + (-13.2 + 5.46i)T + (594. - 594. i)T^{2} \)
31 \( 1 - 32.3iT - 961T^{2} \)
37 \( 1 + (-13.3 + 32.2i)T + (-968. - 968. i)T^{2} \)
41 \( 1 + (-41.1 + 41.1i)T - 1.68e3iT^{2} \)
43 \( 1 + (23.3 - 56.3i)T + (-1.30e3 - 1.30e3i)T^{2} \)
47 \( 1 + 12.8T + 2.20e3T^{2} \)
53 \( 1 + (40.9 + 16.9i)T + (1.98e3 + 1.98e3i)T^{2} \)
59 \( 1 + (52.4 + 21.7i)T + (2.46e3 + 2.46e3i)T^{2} \)
61 \( 1 + (-13.9 + 5.77i)T + (2.63e3 - 2.63e3i)T^{2} \)
67 \( 1 + (41.9 + 101. i)T + (-3.17e3 + 3.17e3i)T^{2} \)
71 \( 1 + (19.4 - 19.4i)T - 5.04e3iT^{2} \)
73 \( 1 + (-5.72 + 5.72i)T - 5.32e3iT^{2} \)
79 \( 1 - 100.T + 6.24e3T^{2} \)
83 \( 1 + (-41.3 + 17.1i)T + (4.87e3 - 4.87e3i)T^{2} \)
89 \( 1 + (37.4 + 37.4i)T + 7.92e3iT^{2} \)
97 \( 1 - 43.9T + 9.40e3T^{2} \)
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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

−12.38653332363437063707989272255, −10.42247358364188070927399877517, −9.386989775898068024398143065755, −8.295554233936225775490629553654, −7.71614914676963664979764462182, −6.31706383654432067558716547776, −5.67038817163110255188265530188, −4.78850457824965040500841467791, −1.70341213226760434482979217637, −0.45622686754476470867253741217, 2.57504961734575125284291455095, 3.58837444724877245516381573543, 4.80118541103313862490696421467, 6.10285979141864385622372714275, 7.37883558457064411784095187712, 9.312148730049491666391220636443, 9.781338837865163003258618000101, 10.44875258988556512420247356905, 11.32319069016595688630836739224, 11.69777750190131239806543220179

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