Properties

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

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 7-s − 9-s − 2·23-s − 25-s + 49-s − 63-s + 2·71-s + 2·79-s + 81-s − 2·113-s + ⋯
L(s)  = 1  + 7-s − 9-s − 2·23-s − 25-s + 49-s − 63-s + 2·71-s + 2·79-s + 81-s − 2·113-s + ⋯

Functional equation

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

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(224\)    =    \(2^{5} \cdot 7\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(0\)
character  :  $\chi_{224} (209, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 224,\ (\ :0),\ 1)$
$L(\frac{1}{2})$  $\approx$  $0.7266038497$
$L(\frac12)$  $\approx$  $0.7266038497$
$L(1)$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;7\}$, \(F_p\) is a polynomial of degree 2. If $p \in \{2,\;7\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 \)
7 \( 1 - T \)
good3 \( 1 + T^{2} \)
5 \( 1 + T^{2} \)
11 \( ( 1 - T )( 1 + T ) \)
13 \( 1 + T^{2} \)
17 \( ( 1 - T )( 1 + T ) \)
19 \( 1 + T^{2} \)
23 \( ( 1 + T )^{2} \)
29 \( ( 1 - T )( 1 + T ) \)
31 \( ( 1 - T )( 1 + T ) \)
37 \( ( 1 - T )( 1 + T ) \)
41 \( ( 1 - T )( 1 + T ) \)
43 \( ( 1 - T )( 1 + T ) \)
47 \( ( 1 - T )( 1 + T ) \)
53 \( ( 1 - T )( 1 + T ) \)
59 \( 1 + T^{2} \)
61 \( 1 + T^{2} \)
67 \( ( 1 - T )( 1 + T ) \)
71 \( ( 1 - T )^{2} \)
73 \( ( 1 - T )( 1 + T ) \)
79 \( ( 1 - T )^{2} \)
83 \( 1 + T^{2} \)
89 \( ( 1 - T )( 1 + T ) \)
97 \( ( 1 - T )( 1 + T ) \)
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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

−12.22369903725000176271682439712, −11.58836941877609302991849398627, −10.69844401423555597352562836766, −9.562484600326469358309217586395, −8.370993614097800613301780281956, −7.76810251622520307131534426357, −6.22032254060822648294042949415, −5.24138774305761413651259165297, −3.89328530717634633460037294551, −2.16883543425635289528621359563, 2.16883543425635289528621359563, 3.89328530717634633460037294551, 5.24138774305761413651259165297, 6.22032254060822648294042949415, 7.76810251622520307131534426357, 8.370993614097800613301780281956, 9.562484600326469358309217586395, 10.69844401423555597352562836766, 11.58836941877609302991849398627, 12.22369903725000176271682439712

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