Properties

Degree 2
Conductor $ 2^{5} \cdot 7 $
Sign $-0.751 - 0.660i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.80 + 0.860i)2-s + (1.98 + 4.78i)3-s + (2.51 − 3.10i)4-s + (1.28 − 3.11i)5-s + (−7.69 − 6.93i)6-s + (1.87 − 1.87i)7-s + (−1.87 + 7.77i)8-s + (−12.5 + 12.5i)9-s + (0.351 + 6.72i)10-s + (−2.68 + 6.47i)11-s + (19.8 + 5.88i)12-s + (4.80 + 11.5i)13-s + (−1.76 + 4.98i)14-s + 17.4·15-s + (−3.31 − 15.6i)16-s + 32.2i·17-s + ⋯
L(s)  = 1  + (−0.902 + 0.430i)2-s + (0.660 + 1.59i)3-s + (0.629 − 0.776i)4-s + (0.257 − 0.622i)5-s + (−1.28 − 1.15i)6-s + (0.267 − 0.267i)7-s + (−0.233 + 0.972i)8-s + (−1.39 + 1.39i)9-s + (0.0351 + 0.672i)10-s + (−0.243 + 0.588i)11-s + (1.65 + 0.490i)12-s + (0.369 + 0.891i)13-s + (−0.126 + 0.356i)14-s + 1.16·15-s + (−0.207 − 0.978i)16-s + 1.89i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.751 - 0.660i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.751 - 0.660i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(224\)    =    \(2^{5} \cdot 7\)
\( \varepsilon \)  =  $-0.751 - 0.660i$
motivic weight  =  \(2\)
character  :  $\chi_{224} (43, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 224,\ (\ :1),\ -0.751 - 0.660i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.432323 + 1.14710i\)
\(L(\frac12)\)  \(\approx\)  \(0.432323 + 1.14710i\)
\(L(2)\)   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 \( 1 + (1.80 - 0.860i)T \)
7 \( 1 + (-1.87 + 1.87i)T \)
good3 \( 1 + (-1.98 - 4.78i)T + (-6.36 + 6.36i)T^{2} \)
5 \( 1 + (-1.28 + 3.11i)T + (-17.6 - 17.6i)T^{2} \)
11 \( 1 + (2.68 - 6.47i)T + (-85.5 - 85.5i)T^{2} \)
13 \( 1 + (-4.80 - 11.5i)T + (-119. + 119. i)T^{2} \)
17 \( 1 - 32.2iT - 289T^{2} \)
19 \( 1 + (2.14 - 0.889i)T + (255. - 255. i)T^{2} \)
23 \( 1 + (20.4 + 20.4i)T + 529iT^{2} \)
29 \( 1 + (-6.33 + 2.62i)T + (594. - 594. i)T^{2} \)
31 \( 1 - 2.54iT - 961T^{2} \)
37 \( 1 + (-2.41 + 5.82i)T + (-968. - 968. i)T^{2} \)
41 \( 1 + (23.5 - 23.5i)T - 1.68e3iT^{2} \)
43 \( 1 + (-22.2 + 53.6i)T + (-1.30e3 - 1.30e3i)T^{2} \)
47 \( 1 - 79.1T + 2.20e3T^{2} \)
53 \( 1 + (4.48 + 1.85i)T + (1.98e3 + 1.98e3i)T^{2} \)
59 \( 1 + (7.84 + 3.25i)T + (2.46e3 + 2.46e3i)T^{2} \)
61 \( 1 + (25.7 - 10.6i)T + (2.63e3 - 2.63e3i)T^{2} \)
67 \( 1 + (43.4 + 104. i)T + (-3.17e3 + 3.17e3i)T^{2} \)
71 \( 1 + (44.8 - 44.8i)T - 5.04e3iT^{2} \)
73 \( 1 + (-90.7 + 90.7i)T - 5.32e3iT^{2} \)
79 \( 1 - 133.T + 6.24e3T^{2} \)
83 \( 1 + (-76.7 + 31.7i)T + (4.87e3 - 4.87e3i)T^{2} \)
89 \( 1 + (42.5 + 42.5i)T + 7.92e3iT^{2} \)
97 \( 1 - 158.T + 9.40e3T^{2} \)
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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

−12.18973531512220908811785543242, −10.68100931313653480365112265135, −10.42445597016854460256623394088, −9.295041596665617200737587372386, −8.736065877751245983785735957501, −7.88601007291602609952184692393, −6.24120821053280019983154076563, −4.95821976795176793066031548310, −3.96730025075257758034827801541, −1.98972725987403120121097449519, 0.842430079714724255201699867645, 2.38083707063204361735461320056, 3.12163908033787600666279204512, 5.89964397719683850322892838839, 6.99266593529483197853130409007, 7.73572195579245852245287172113, 8.506563199527005078087163660538, 9.516936310017558243625975005132, 10.78321612246198385826435579337, 11.71106349844211543019586870150

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