Properties

Degree 2
Conductor $ 2^{5} \cdot 7 $
Sign $0.890 - 0.454i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.55 + 1.26i)2-s + (−1.89 − 4.58i)3-s + (0.810 − 3.91i)4-s + (−2.09 + 5.04i)5-s + (8.73 + 4.71i)6-s + (−1.87 + 1.87i)7-s + (3.68 + 7.09i)8-s + (−11.0 + 11.0i)9-s + (−3.13 − 10.4i)10-s + (0.560 − 1.35i)11-s + (−19.5 + 3.72i)12-s + (0.285 + 0.688i)13-s + (0.539 − 5.26i)14-s + 27.1·15-s + (−14.6 − 6.35i)16-s + 2.08i·17-s + ⋯
L(s)  = 1  + (−0.775 + 0.631i)2-s + (−0.633 − 1.52i)3-s + (0.202 − 0.979i)4-s + (−0.418 + 1.00i)5-s + (1.45 + 0.785i)6-s + (−0.267 + 0.267i)7-s + (0.461 + 0.887i)8-s + (−1.22 + 1.22i)9-s + (−0.313 − 1.04i)10-s + (0.0509 − 0.123i)11-s + (−1.62 + 0.310i)12-s + (0.0219 + 0.0529i)13-s + (0.0385 − 0.375i)14-s + 1.80·15-s + (−0.917 − 0.396i)16-s + 0.122i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.890 - 0.454i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.890 - 0.454i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(224\)    =    \(2^{5} \cdot 7\)
\( \varepsilon \)  =  $0.890 - 0.454i$
motivic weight  =  \(2\)
character  :  $\chi_{224} (43, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 224,\ (\ :1),\ 0.890 - 0.454i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.669418 + 0.160788i\)
\(L(\frac12)\)  \(\approx\)  \(0.669418 + 0.160788i\)
\(L(2)\)   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 \( 1 + (1.55 - 1.26i)T \)
7 \( 1 + (1.87 - 1.87i)T \)
good3 \( 1 + (1.89 + 4.58i)T + (-6.36 + 6.36i)T^{2} \)
5 \( 1 + (2.09 - 5.04i)T + (-17.6 - 17.6i)T^{2} \)
11 \( 1 + (-0.560 + 1.35i)T + (-85.5 - 85.5i)T^{2} \)
13 \( 1 + (-0.285 - 0.688i)T + (-119. + 119. i)T^{2} \)
17 \( 1 - 2.08iT - 289T^{2} \)
19 \( 1 + (-26.8 + 11.1i)T + (255. - 255. i)T^{2} \)
23 \( 1 + (-26.5 - 26.5i)T + 529iT^{2} \)
29 \( 1 + (-22.2 + 9.19i)T + (594. - 594. i)T^{2} \)
31 \( 1 + 22.4iT - 961T^{2} \)
37 \( 1 + (-1.44 + 3.49i)T + (-968. - 968. i)T^{2} \)
41 \( 1 + (45.8 - 45.8i)T - 1.68e3iT^{2} \)
43 \( 1 + (13.3 - 32.2i)T + (-1.30e3 - 1.30e3i)T^{2} \)
47 \( 1 - 75.2T + 2.20e3T^{2} \)
53 \( 1 + (-96.7 - 40.0i)T + (1.98e3 + 1.98e3i)T^{2} \)
59 \( 1 + (10.3 + 4.26i)T + (2.46e3 + 2.46e3i)T^{2} \)
61 \( 1 + (-23.0 + 9.55i)T + (2.63e3 - 2.63e3i)T^{2} \)
67 \( 1 + (1.14 + 2.77i)T + (-3.17e3 + 3.17e3i)T^{2} \)
71 \( 1 + (71.6 - 71.6i)T - 5.04e3iT^{2} \)
73 \( 1 + (70.1 - 70.1i)T - 5.32e3iT^{2} \)
79 \( 1 + 30.4T + 6.24e3T^{2} \)
83 \( 1 + (-128. + 53.1i)T + (4.87e3 - 4.87e3i)T^{2} \)
89 \( 1 + (47.6 + 47.6i)T + 7.92e3iT^{2} \)
97 \( 1 - 60.2T + 9.40e3T^{2} \)
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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

−11.62430300511150603487327141822, −11.46576434895504772158427644405, −10.20077057441343443938224691375, −8.927015590908177897365365044150, −7.62080041951366029576377211489, −7.18614035756235087798271874528, −6.34650096437060873775780515546, −5.38212029770167431256547862984, −2.76989389112787481542652825296, −1.05620331501038539457841881156, 0.69208207062841192763434011712, 3.30389741730918156453888953507, 4.35690121218840131174322537319, 5.28715011908153098286499989763, 7.05943658195608890906693177526, 8.555852560640576401620479903982, 9.137240767252067279380362108631, 10.18667570244996022210736949930, 10.66875663160181556358456755733, 11.89522972600929366380581416600

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