Properties

Degree 2
Conductor $ 2^{5} \cdot 7 $
Sign $-0.707 + 0.707i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 3.85i·3-s + 0.490·5-s − 2.64i·7-s − 5.87·9-s − 15.5i·11-s + 3.50·13-s − 1.89i·15-s − 24.1·17-s − 3.56i·19-s − 10.2·21-s + 19.5i·23-s − 24.7·25-s − 12.0i·27-s − 10.9·29-s − 21.1i·31-s + ⋯
L(s)  = 1  − 1.28i·3-s + 0.0980·5-s − 0.377i·7-s − 0.652·9-s − 1.41i·11-s + 0.269·13-s − 0.126i·15-s − 1.42·17-s − 0.187i·19-s − 0.485·21-s + 0.851i·23-s − 0.990·25-s − 0.446i·27-s − 0.378·29-s − 0.683i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(224\)    =    \(2^{5} \cdot 7\)
\( \varepsilon \)  =  $-0.707 + 0.707i$
motivic weight  =  \(2\)
character  :  $\chi_{224} (127, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 224,\ (\ :1),\ -0.707 + 0.707i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.513126 - 1.23879i\)
\(L(\frac12)\)  \(\approx\)  \(0.513126 - 1.23879i\)
\(L(2)\)   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 \( 1 + 2.64iT \)
good3 \( 1 + 3.85iT - 9T^{2} \)
5 \( 1 - 0.490T + 25T^{2} \)
11 \( 1 + 15.5iT - 121T^{2} \)
13 \( 1 - 3.50T + 169T^{2} \)
17 \( 1 + 24.1T + 289T^{2} \)
19 \( 1 + 3.56iT - 361T^{2} \)
23 \( 1 - 19.5iT - 529T^{2} \)
29 \( 1 + 10.9T + 841T^{2} \)
31 \( 1 + 21.1iT - 961T^{2} \)
37 \( 1 - 58.4T + 1.36e3T^{2} \)
41 \( 1 - 54.1T + 1.68e3T^{2} \)
43 \( 1 - 35.6iT - 1.84e3T^{2} \)
47 \( 1 + 64.2iT - 2.20e3T^{2} \)
53 \( 1 - 87.4T + 2.80e3T^{2} \)
59 \( 1 + 66.6iT - 3.48e3T^{2} \)
61 \( 1 - 16.8T + 3.72e3T^{2} \)
67 \( 1 + 21.2iT - 4.48e3T^{2} \)
71 \( 1 - 64.2iT - 5.04e3T^{2} \)
73 \( 1 - 99.4T + 5.32e3T^{2} \)
79 \( 1 - 139. iT - 6.24e3T^{2} \)
83 \( 1 + 6.03iT - 6.88e3T^{2} \)
89 \( 1 + 23.9T + 7.92e3T^{2} \)
97 \( 1 - 171.T + 9.40e3T^{2} \)
show more
show less
\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−11.52615769514746196412449782562, −11.11587529782133437505429868712, −9.609321718445248114625443464379, −8.454236923603935112302956759997, −7.62119354533130778962109115385, −6.57300805900397042189837446874, −5.75489480847132335759953256441, −3.95141754454743559180683873301, −2.30104743066743134171992057560, −0.73647826206391503167501006961, 2.32006634762498395378840636344, 4.06514139371546035048262565926, 4.74812777397152279311185585249, 6.08116368829768638684368408717, 7.39442163451011055178752556854, 8.816518537028434450896906404282, 9.522029217894562747299043217774, 10.36277184031552392153539748149, 11.20558793723990992699362468396, 12.33349051749343309196921655477

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