Properties

Degree 4
Conductor $ 2^{7} \cdot 5 $
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  − 2·3-s − 3·5-s + 2·7-s − 2·9-s + 8·13-s + 6·15-s − 4·17-s − 4·19-s − 4·21-s + 6·23-s + 2·25-s + 10·27-s − 4·29-s − 4·31-s − 6·35-s − 16·39-s + 16·41-s − 10·43-s + 6·45-s − 6·47-s − 10·49-s + 8·51-s + 8·53-s + 8·57-s + 12·59-s − 8·61-s − 4·63-s + ⋯
L(s)  = 1  − 1.15·3-s − 1.34·5-s + 0.755·7-s − 2/3·9-s + 2.21·13-s + 1.54·15-s − 0.970·17-s − 0.917·19-s − 0.872·21-s + 1.25·23-s + 2/5·25-s + 1.92·27-s − 0.742·29-s − 0.718·31-s − 1.01·35-s − 2.56·39-s + 2.49·41-s − 1.52·43-s + 0.894·45-s − 0.875·47-s − 1.42·49-s + 1.12·51-s + 1.09·53-s + 1.05·57-s + 1.56·59-s − 1.02·61-s − 0.503·63-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(640\)    =    \(2^{7} \cdot 5\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{640} (1, \cdot )$
Sato-Tate  :  $N(G_{1,3})$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 640,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $0.3085697573$
$L(\frac12)$  $\approx$  $0.3085697573$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

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

−19.6035760866, −19.1887245783, −18.1271877078, −18.1183629461, −17.3081878781, −16.7385636699, −16.0995603957, −15.7488207479, −14.7919331095, −14.5765256398, −13.2768755254, −13.0023664158, −11.7666126827, −11.508544532, −10.9076921437, −10.834739503, −8.95538623117, −8.55221755078, −7.77199473906, −6.57891116466, −5.87146418849, −4.78130792718, −3.67478222653, 3.67478222653, 4.78130792718, 5.87146418849, 6.57891116466, 7.77199473906, 8.55221755078, 8.95538623117, 10.834739503, 10.9076921437, 11.508544532, 11.7666126827, 13.0023664158, 13.2768755254, 14.5765256398, 14.7919331095, 15.7488207479, 16.0995603957, 16.7385636699, 17.3081878781, 18.1183629461, 18.1271877078, 19.1887245783, 19.6035760866

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