Properties

Degree 2
Conductor $ 2^{5} \cdot 7 $
Sign $0.758 - 0.651i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.455 + 0.788i)3-s + (−3.17 − 5.49i)5-s + (−3.79 + 5.88i)7-s + (4.08 + 7.07i)9-s + (11.4 + 6.60i)11-s + 19.4·13-s + 5.77·15-s + (13.7 + 7.96i)17-s + (−8.22 − 14.2i)19-s + (−2.91 − 5.67i)21-s + (11.9 + 20.7i)23-s + (−7.62 + 13.2i)25-s − 15.6·27-s + 16.6i·29-s + (11.1 + 6.42i)31-s + ⋯
L(s)  = 1  + (−0.151 + 0.262i)3-s + (−0.634 − 1.09i)5-s + (−0.541 + 0.840i)7-s + (0.453 + 0.786i)9-s + (1.04 + 0.600i)11-s + 1.49·13-s + 0.385·15-s + (0.811 + 0.468i)17-s + (−0.433 − 0.750i)19-s + (−0.138 − 0.270i)21-s + (0.520 + 0.900i)23-s + (−0.305 + 0.528i)25-s − 0.579·27-s + 0.574i·29-s + (0.359 + 0.207i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.758 - 0.651i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.758 - 0.651i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(224\)    =    \(2^{5} \cdot 7\)
\( \varepsilon \)  =  $0.758 - 0.651i$
motivic weight  =  \(2\)
character  :  $\chi_{224} (17, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 224,\ (\ :1),\ 0.758 - 0.651i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(1.28857 + 0.477359i\)
\(L(\frac12)\)  \(\approx\)  \(1.28857 + 0.477359i\)
\(L(2)\)   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 \( 1 + (3.79 - 5.88i)T \)
good3 \( 1 + (0.455 - 0.788i)T + (-4.5 - 7.79i)T^{2} \)
5 \( 1 + (3.17 + 5.49i)T + (-12.5 + 21.6i)T^{2} \)
11 \( 1 + (-11.4 - 6.60i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 - 19.4T + 169T^{2} \)
17 \( 1 + (-13.7 - 7.96i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (8.22 + 14.2i)T + (-180.5 + 312. i)T^{2} \)
23 \( 1 + (-11.9 - 20.7i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 - 16.6iT - 841T^{2} \)
31 \( 1 + (-11.1 - 6.42i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (41.1 - 23.7i)T + (684.5 - 1.18e3i)T^{2} \)
41 \( 1 - 6.49iT - 1.68e3T^{2} \)
43 \( 1 + 33.2iT - 1.84e3T^{2} \)
47 \( 1 + (-18.9 + 10.9i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (-32.2 - 18.5i)T + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (27.3 - 47.3i)T + (-1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-5.12 - 8.87i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (14.8 + 8.56i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 32.0T + 5.04e3T^{2} \)
73 \( 1 + (-92.8 - 53.5i)T + (2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (29.1 + 50.4i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + 36.3T + 6.88e3T^{2} \)
89 \( 1 + (-0.929 + 0.536i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + 169. iT - 9.40e3T^{2} \)
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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

−12.18556762364814743705584812262, −11.32940707936565206434756653607, −10.15113359503229068144383116703, −8.981257278877573815474484621452, −8.495517535869171606665913980321, −7.10042591584463384092998119582, −5.78693628790634758490398309205, −4.69589234092415515289800691328, −3.60582200181263239844196852423, −1.44201415134418423082223040576, 0.926952523146857684531643855841, 3.41355546012049153465906734028, 3.91081261008206662942382479265, 6.21946360109798664047480490156, 6.67247066808077269357136655314, 7.71582118357570915016813621554, 8.981993207156088997534565056914, 10.18093899265840270193388172369, 10.99127581119009304852619784401, 11.80260475259280279920940568585

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