Properties

Degree 2
Conductor $ 2^{5} \cdot 7 $
Sign $1$
Motivic weight 2
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.82·3-s + 8.48·5-s + 7·7-s − 0.999·9-s − 25.4·13-s + 24·15-s − 8.48·19-s + 19.7·21-s + 10·23-s + 46.9·25-s − 28.2·27-s + 59.3·35-s − 72·39-s − 8.48·45-s + 49·49-s − 24·57-s − 76.3·59-s + 8.48·61-s − 6.99·63-s − 215.·65-s + 28.2·69-s + 110·71-s + 132.·75-s − 130·79-s − 71.0·81-s + 25.4·83-s − 178.·91-s + ⋯
L(s)  = 1  + 0.942·3-s + 1.69·5-s + 7-s − 0.111·9-s − 1.95·13-s + 1.60·15-s − 0.446·19-s + 0.942·21-s + 0.434·23-s + 1.87·25-s − 1.04·27-s + 1.69·35-s − 1.84·39-s − 0.188·45-s + 0.999·49-s − 0.421·57-s − 1.29·59-s + 0.139·61-s − 0.111·63-s − 3.32·65-s + 0.409·69-s + 1.54·71-s + 1.77·75-s − 1.64·79-s − 0.876·81-s + 0.306·83-s − 1.95·91-s + ⋯

Functional equation

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

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(224\)    =    \(2^{5} \cdot 7\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(2\)
character  :  $\chi_{224} (209, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 224,\ (\ :1),\ 1)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(2.63555\)
\(L(\frac12)\)  \(\approx\)  \(2.63555\)
\(L(2)\)   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 \( 1 - 7T \)
good3 \( 1 - 2.82T + 9T^{2} \)
5 \( 1 - 8.48T + 25T^{2} \)
11 \( 1 - 121T^{2} \)
13 \( 1 + 25.4T + 169T^{2} \)
17 \( 1 - 289T^{2} \)
19 \( 1 + 8.48T + 361T^{2} \)
23 \( 1 - 10T + 529T^{2} \)
29 \( 1 - 841T^{2} \)
31 \( 1 - 961T^{2} \)
37 \( 1 - 1.36e3T^{2} \)
41 \( 1 - 1.68e3T^{2} \)
43 \( 1 - 1.84e3T^{2} \)
47 \( 1 - 2.20e3T^{2} \)
53 \( 1 - 2.80e3T^{2} \)
59 \( 1 + 76.3T + 3.48e3T^{2} \)
61 \( 1 - 8.48T + 3.72e3T^{2} \)
67 \( 1 - 4.48e3T^{2} \)
71 \( 1 - 110T + 5.04e3T^{2} \)
73 \( 1 - 5.32e3T^{2} \)
79 \( 1 + 130T + 6.24e3T^{2} \)
83 \( 1 - 25.4T + 6.88e3T^{2} \)
89 \( 1 - 7.92e3T^{2} \)
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

−12.17908525126606484546904293654, −10.86937113443377711471404444849, −9.835743042178350734614050249043, −9.204737382735169592152918810480, −8.192562701493007242981464210286, −7.08435692803129348903373701827, −5.67167583127100728382666263997, −4.74719880245968070293012242350, −2.70912996390944121597786294158, −1.91221612892844471889225483613, 1.91221612892844471889225483613, 2.70912996390944121597786294158, 4.74719880245968070293012242350, 5.67167583127100728382666263997, 7.08435692803129348903373701827, 8.192562701493007242981464210286, 9.204737382735169592152918810480, 9.835743042178350734614050249043, 10.86937113443377711471404444849, 12.17908525126606484546904293654

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