Properties

Degree 2
Conductor $ 2^{5} \cdot 7 $
Sign $0.872 + 0.487i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.125 + 1.99i)2-s + (−3.96 − 0.5i)4-s + (4.94 − 4.94i)7-s + (1.49 − 7.85i)8-s + (−6.36 − 6.36i)9-s + (−5.22 − 12.6i)11-s + (9.26 + 10.5i)14-s + (15.5 + 3.96i)16-s + (13.5 − 11.9i)18-s + (25.8 − 8.84i)22-s + (12.1 + 12.1i)23-s + (17.6 − 17.6i)25-s + (−22.1 + 17.1i)28-s + (22.1 − 53.5i)29-s + (−9.86 + 30.4i)32-s + ⋯
L(s)  = 1  + (−0.0626 + 0.998i)2-s + (−0.992 − 0.125i)4-s + (0.707 − 0.707i)7-s + (0.186 − 0.982i)8-s + (−0.707 − 0.707i)9-s + (−0.474 − 1.14i)11-s + (0.661 + 0.750i)14-s + (0.968 + 0.248i)16-s + (0.750 − 0.661i)18-s + (1.17 − 0.401i)22-s + (0.528 + 0.528i)23-s + (0.707 − 0.707i)25-s + (−0.789 + 0.613i)28-s + (0.765 − 1.84i)29-s + (−0.308 + 0.951i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.872 + 0.487i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.872 + 0.487i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(224\)    =    \(2^{5} \cdot 7\)
\( \varepsilon \)  =  $0.872 + 0.487i$
motivic weight  =  \(2\)
character  :  $\chi_{224} (13, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 224,\ (\ :1),\ 0.872 + 0.487i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(1.07564 - 0.280202i\)
\(L(\frac12)\)  \(\approx\)  \(1.07564 - 0.280202i\)
\(L(2)\)   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 \( 1 + (0.125 - 1.99i)T \)
7 \( 1 + (-4.94 + 4.94i)T \)
good3 \( 1 + (6.36 + 6.36i)T^{2} \)
5 \( 1 + (-17.6 + 17.6i)T^{2} \)
11 \( 1 + (5.22 + 12.6i)T + (-85.5 + 85.5i)T^{2} \)
13 \( 1 + (-119. - 119. i)T^{2} \)
17 \( 1 + 289T^{2} \)
19 \( 1 + (-255. - 255. i)T^{2} \)
23 \( 1 + (-12.1 - 12.1i)T + 529iT^{2} \)
29 \( 1 + (-22.1 + 53.5i)T + (-594. - 594. i)T^{2} \)
31 \( 1 - 961T^{2} \)
37 \( 1 + (67.6 - 28.0i)T + (968. - 968. i)T^{2} \)
41 \( 1 - 1.68e3iT^{2} \)
43 \( 1 + (-13.9 - 33.6i)T + (-1.30e3 + 1.30e3i)T^{2} \)
47 \( 1 + 2.20e3T^{2} \)
53 \( 1 + (36.5 + 88.2i)T + (-1.98e3 + 1.98e3i)T^{2} \)
59 \( 1 + (-2.46e3 + 2.46e3i)T^{2} \)
61 \( 1 + (2.63e3 + 2.63e3i)T^{2} \)
67 \( 1 + (39.7 - 95.9i)T + (-3.17e3 - 3.17e3i)T^{2} \)
71 \( 1 + (-59.8 + 59.8i)T - 5.04e3iT^{2} \)
73 \( 1 - 5.32e3iT^{2} \)
79 \( 1 + 23.3iT - 6.24e3T^{2} \)
83 \( 1 + (-4.87e3 - 4.87e3i)T^{2} \)
89 \( 1 + 7.92e3iT^{2} \)
97 \( 1 - 9.40e3T^{2} \)
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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

−11.93666597528484827744846432791, −10.91900711677508792985880485279, −9.861672426173572942774274440298, −8.599155014470524383643794810041, −8.090155390811512978557883958173, −6.82846255341272509802982864586, −5.84808145296909447629546563805, −4.73764933412013642423289347243, −3.37365948929583932237905658919, −0.64497943976343179949334934172, 1.81881636968344228230828217856, 2.95829285055756063958438994847, 4.76365539647367091052042110830, 5.35459421681946259652703290606, 7.31836950272198574316407771636, 8.486592388022060588268116775284, 9.130068978928691004382501489876, 10.52995605045434347675397552454, 10.97924902398272152857633817516, 12.20543728392601134962346700516

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