Properties

Degree 2
Conductor $ 2^{5} \cdot 7 $
Sign $0.863 + 0.504i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−3.45 − 1.99i)3-s + (−7.80 + 4.50i)5-s + (5.54 + 4.26i)7-s + (3.44 + 5.96i)9-s + (8.28 − 14.3i)11-s + 0.446i·13-s + 35.9·15-s + (6.02 + 3.47i)17-s + (11.0 − 6.37i)19-s + (−10.6 − 25.7i)21-s + (−13.2 − 23.0i)23-s + (28.1 − 48.7i)25-s + 8.43i·27-s + 26.4·29-s + (21.7 + 12.5i)31-s + ⋯
L(s)  = 1  + (−1.15 − 0.664i)3-s + (−1.56 + 0.901i)5-s + (0.792 + 0.609i)7-s + (0.382 + 0.662i)9-s + (0.752 − 1.30i)11-s + 0.0343i·13-s + 2.39·15-s + (0.354 + 0.204i)17-s + (0.580 − 0.335i)19-s + (−0.507 − 1.22i)21-s + (−0.577 − 1.00i)23-s + (1.12 − 1.95i)25-s + 0.312i·27-s + 0.912·29-s + (0.702 + 0.405i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.863 + 0.504i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.863 + 0.504i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(224\)    =    \(2^{5} \cdot 7\)
\( \varepsilon \)  =  $0.863 + 0.504i$
motivic weight  =  \(2\)
character  :  $\chi_{224} (129, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 224,\ (\ :1),\ 0.863 + 0.504i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.792487 - 0.214538i\)
\(L(\frac12)\)  \(\approx\)  \(0.792487 - 0.214538i\)
\(L(2)\)   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 \( 1 \)
7 \( 1 + (-5.54 - 4.26i)T \)
good3 \( 1 + (3.45 + 1.99i)T + (4.5 + 7.79i)T^{2} \)
5 \( 1 + (7.80 - 4.50i)T + (12.5 - 21.6i)T^{2} \)
11 \( 1 + (-8.28 + 14.3i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 - 0.446iT - 169T^{2} \)
17 \( 1 + (-6.02 - 3.47i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (-11.0 + 6.37i)T + (180.5 - 312. i)T^{2} \)
23 \( 1 + (13.2 + 23.0i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 - 26.4T + 841T^{2} \)
31 \( 1 + (-21.7 - 12.5i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (-31.6 - 54.9i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 - 0.519iT - 1.68e3T^{2} \)
43 \( 1 - 25.5T + 1.84e3T^{2} \)
47 \( 1 + (-59.4 + 34.3i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (-3.58 + 6.20i)T + (-1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (65.3 + 37.7i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-39.8 + 23.0i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-21.4 + 37.0i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 60.0T + 5.04e3T^{2} \)
73 \( 1 + (40.5 + 23.3i)T + (2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (27.1 + 47.0i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 - 11.4iT - 6.88e3T^{2} \)
89 \( 1 + (-53.1 + 30.7i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 - 20.3iT - 9.40e3T^{2} \)
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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

−11.81132931080940571340178940401, −11.32760163426694028726950626753, −10.55329371342683071655749357936, −8.626132201601506312398942675741, −7.86657414142194974333645031895, −6.74901990187981550849434105912, −5.98269622912312630141361990947, −4.54575787335214858590484041431, −3.10491990161699864803769981103, −0.76857155342408799668170881777, 0.967093651953591820472313406476, 4.12792880479239847312567048013, 4.43801340444367759120504086419, 5.56173113742598402622263163518, 7.27630437040419859651565889360, 7.930611402719717203531445916185, 9.298672825076333519751949475556, 10.35893532885239608524896236197, 11.43386643034969002995300220259, 11.85748378293746483143999143668

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