Properties

Degree 2
Conductor $ 2^{5} \cdot 7 $
Sign $0.993 - 0.116i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.99 − 0.125i)2-s + (3.96 − 0.5i)4-s + (−4.94 + 4.94i)7-s + (7.85 − 1.49i)8-s + (6.36 + 6.36i)9-s + (20.1 − 8.36i)11-s + (−9.26 + 10.5i)14-s + (15.5 − 3.96i)16-s + (13.5 + 11.9i)18-s + (39.2 − 19.2i)22-s + (−30.1 − 30.1i)23-s + (−17.6 + 17.6i)25-s + (−17.1 + 22.1i)28-s + (−37.1 − 15.3i)29-s + (30.4 − 9.86i)32-s + ⋯
L(s)  = 1  + (0.998 − 0.0626i)2-s + (0.992 − 0.125i)4-s + (−0.707 + 0.707i)7-s + (0.982 − 0.186i)8-s + (0.707 + 0.707i)9-s + (1.83 − 0.760i)11-s + (−0.661 + 0.750i)14-s + (0.968 − 0.248i)16-s + (0.750 + 0.661i)18-s + (1.78 − 0.873i)22-s + (−1.31 − 1.31i)23-s + (−0.707 + 0.707i)25-s + (−0.613 + 0.789i)28-s + (−1.28 − 0.530i)29-s + (0.951 − 0.308i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 - 0.116i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.993 - 0.116i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(224\)    =    \(2^{5} \cdot 7\)
\( \varepsilon \)  =  $0.993 - 0.116i$
motivic weight  =  \(2\)
character  :  $\chi_{224} (125, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 224,\ (\ :1),\ 0.993 - 0.116i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(2.95935 + 0.173275i\)
\(L(\frac12)\)  \(\approx\)  \(2.95935 + 0.173275i\)
\(L(2)\)   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 \( 1 + (-1.99 + 0.125i)T \)
7 \( 1 + (4.94 - 4.94i)T \)
good3 \( 1 + (-6.36 - 6.36i)T^{2} \)
5 \( 1 + (17.6 - 17.6i)T^{2} \)
11 \( 1 + (-20.1 + 8.36i)T + (85.5 - 85.5i)T^{2} \)
13 \( 1 + (119. + 119. i)T^{2} \)
17 \( 1 + 289T^{2} \)
19 \( 1 + (255. + 255. i)T^{2} \)
23 \( 1 + (30.1 + 30.1i)T + 529iT^{2} \)
29 \( 1 + (37.1 + 15.3i)T + (594. + 594. i)T^{2} \)
31 \( 1 - 961T^{2} \)
37 \( 1 + (-22.7 - 54.8i)T + (-968. + 968. i)T^{2} \)
41 \( 1 - 1.68e3iT^{2} \)
43 \( 1 + (27.0 - 11.2i)T + (1.30e3 - 1.30e3i)T^{2} \)
47 \( 1 + 2.20e3T^{2} \)
53 \( 1 + (32.2 - 13.3i)T + (1.98e3 - 1.98e3i)T^{2} \)
59 \( 1 + (2.46e3 - 2.46e3i)T^{2} \)
61 \( 1 + (-2.63e3 - 2.63e3i)T^{2} \)
67 \( 1 + (123. + 51.0i)T + (3.17e3 + 3.17e3i)T^{2} \)
71 \( 1 + (-59.8 + 59.8i)T - 5.04e3iT^{2} \)
73 \( 1 - 5.32e3iT^{2} \)
79 \( 1 + 156. iT - 6.24e3T^{2} \)
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

−12.04561477290439114224133284919, −11.46259198708528219410919696122, −10.23368407362721243507877525958, −9.238203259395617351042438723795, −7.87765294532610933470056858134, −6.54104878559044742609159850950, −5.95583811693511436875455894673, −4.46925210215706803719103557867, −3.42722133576052370437153103123, −1.84367102421015085516688006269, 1.62362042559376491096866370713, 3.76044708268531624241938622400, 4.10451022398143114593431585095, 5.91759189732730971550164821271, 6.79486659562344940567949169309, 7.51111956447269080178241672212, 9.382934026044040172077716818189, 10.02418169223078768539808771467, 11.39126843555613096338080271050, 12.17807012134921267534680871259

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