Properties

Degree 4
Conductor $ 2^{6} \cdot 7 $
Sign $1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 2·3-s + 4-s − 2·5-s + 2·6-s + 7-s − 8-s − 2·9-s + 2·10-s − 2·12-s + 2·13-s − 14-s + 4·15-s + 16-s + 8·17-s + 2·18-s + 2·19-s − 2·20-s − 2·21-s + 2·24-s − 6·25-s − 2·26-s + 10·27-s + 28-s − 16·29-s − 4·30-s − 4·31-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.15·3-s + 1/2·4-s − 0.894·5-s + 0.816·6-s + 0.377·7-s − 0.353·8-s − 2/3·9-s + 0.632·10-s − 0.577·12-s + 0.554·13-s − 0.267·14-s + 1.03·15-s + 1/4·16-s + 1.94·17-s + 0.471·18-s + 0.458·19-s − 0.447·20-s − 0.436·21-s + 0.408·24-s − 6/5·25-s − 0.392·26-s + 1.92·27-s + 0.188·28-s − 2.97·29-s − 0.730·30-s − 0.718·31-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(448\)    =    \(2^{6} \cdot 7\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{448} (1, \cdot )$
Sato-Tate  :  $N(G_{1,3})$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 448,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $0.2164663601$
$L(\frac12)$  $\approx$  $0.2164663601$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;7\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;7\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad2$C_1$ \( 1 + T \)
7$C_1$$\times$$C_2$ \( ( 1 - T )( 1 + p T^{2} ) \)
good3$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \)
5$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \)
11$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
17$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \)
19$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + p T^{2} ) \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$$\times$$C_2$ \( ( 1 + 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
31$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
41$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 - 6 T + p T^{2} ) \)
43$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \)
47$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 12 T + p T^{2} ) \)
53$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 - 6 T + p T^{2} ) \)
59$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \)
61$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
67$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
79$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \)
83$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \)
89$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
97$C_2$$\times$$C_2$ \( ( 1 - 18 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−19.5800270198, −19.1887245783, −18.2120541317, −18.1183629461, −17.2128532162, −16.7385636699, −16.3370810014, −15.7488207479, −14.6077730657, −14.5765256398, −13.2768755254, −12.3052609901, −11.7666126827, −11.2313614144, −10.9076921437, −9.76554711946, −8.95538623117, −7.77199473906, −7.57571100089, −5.87146418849, −5.57928681743, −3.67478222653, 3.67478222653, 5.57928681743, 5.87146418849, 7.57571100089, 7.77199473906, 8.95538623117, 9.76554711946, 10.9076921437, 11.2313614144, 11.7666126827, 12.3052609901, 13.2768755254, 14.5765256398, 14.6077730657, 15.7488207479, 16.3370810014, 16.7385636699, 17.2128532162, 18.1183629461, 18.2120541317, 19.1887245783, 19.5800270198

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