Properties

Degree 4
Conductor $ 2^{4} \cdot 5^{2} \cdot 7^{2} $
Sign $1$
Motivic weight 0
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 3-s − 5-s − 6-s + 7-s − 8-s + 9-s − 10-s + 14-s + 15-s − 16-s + 18-s − 21-s − 23-s + 24-s − 2·27-s − 2·29-s + 30-s − 35-s + 40-s − 2·41-s − 42-s + 2·43-s − 45-s − 46-s + 2·47-s + 48-s + ⋯
L(s)  = 1  + 2-s − 3-s − 5-s − 6-s + 7-s − 8-s + 9-s − 10-s + 14-s + 15-s − 16-s + 18-s − 21-s − 23-s + 24-s − 2·27-s − 2·29-s + 30-s − 35-s + 40-s − 2·41-s − 42-s + 2·43-s − 45-s − 46-s + 2·47-s + 48-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 19600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 19600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(19600\)    =    \(2^{4} \cdot 5^{2} \cdot 7^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(0\)
character  :  induced by $\chi_{140} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 19600,\ (\ :0, 0),\ 1)$
$L(\frac{1}{2})$  $\approx$  $0.4070629205$
$L(\frac12)$  $\approx$  $0.4070629205$
$L(1)$   not available
$L(1)$   not available

Euler product

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

−13.72109285787297334621672318409, −12.91953457458712685832370021318, −12.78482575618033582120647928994, −11.89730415182283369245167736241, −11.71208045192582949871815458486, −11.59390338033710325339197060635, −10.67564303383999914924915166349, −10.49131899340443749749830472998, −9.392275456734232582728614569972, −9.186924837552491493319425722300, −8.208014038692492856373607073400, −7.71848071459985778649623621210, −7.29984296400960119785591340565, −6.44825492248817334948738652479, −5.67425283931826133655283997430, −5.42165937945702167287681403977, −4.62317910412855121571655553895, −3.98412937626303021390155283110, −3.69435910234721680945246195208, −2.07304643702320871970159871316, 2.07304643702320871970159871316, 3.69435910234721680945246195208, 3.98412937626303021390155283110, 4.62317910412855121571655553895, 5.42165937945702167287681403977, 5.67425283931826133655283997430, 6.44825492248817334948738652479, 7.29984296400960119785591340565, 7.71848071459985778649623621210, 8.208014038692492856373607073400, 9.186924837552491493319425722300, 9.392275456734232582728614569972, 10.49131899340443749749830472998, 10.67564303383999914924915166349, 11.59390338033710325339197060635, 11.71208045192582949871815458486, 11.89730415182283369245167736241, 12.78482575618033582120647928994, 12.91953457458712685832370021318, 13.72109285787297334621672318409

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