Properties

Degree 4
Conductor $ 2^{2} \cdot 13^{2} $
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 + 4-s − 4·5-s + 9-s + 4·11-s − 2·12-s + 8·15-s + 16-s − 6·17-s + 8·19-s − 4·20-s − 4·23-s + 3·25-s − 2·27-s + 8·29-s − 8·33-s + 36-s − 4·37-s − 6·43-s + 4·44-s − 4·45-s + 16·47-s − 2·48-s − 13·49-s + 12·51-s + 12·53-s − 16·55-s + ⋯
L(s)  = 1  − 1.15·3-s + 1/2·4-s − 1.78·5-s + 1/3·9-s + 1.20·11-s − 0.577·12-s + 2.06·15-s + 1/4·16-s − 1.45·17-s + 1.83·19-s − 0.894·20-s − 0.834·23-s + 3/5·25-s − 0.384·27-s + 1.48·29-s − 1.39·33-s + 1/6·36-s − 0.657·37-s − 0.914·43-s + 0.603·44-s − 0.596·45-s + 2.33·47-s − 0.288·48-s − 1.85·49-s + 1.68·51-s + 1.64·53-s − 2.15·55-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 676 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 676 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(676\)    =    \(2^{2} \cdot 13^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{676} (1, \cdot )$
Sato-Tate  :  $G_{3,3}$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 676,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $0.3201552354$
$L(\frac12)$  $\approx$  $0.3201552354$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

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

−19.6933997517, −19.1444693222, −18.3883990965, −17.518772004, −17.4643953306, −16.3289342707, −16.2105519665, −15.4425168943, −15.2621223361, −14.1081230548, −13.592286388, −12.1499536241, −12.1081086828, −11.4971267324, −11.2148699926, −10.2802576297, −9.25208611998, −8.2891250516, −7.45985560668, −6.76423132328, −5.88908450916, −4.61153453491, −3.64028761626, 3.64028761626, 4.61153453491, 5.88908450916, 6.76423132328, 7.45985560668, 8.2891250516, 9.25208611998, 10.2802576297, 11.2148699926, 11.4971267324, 12.1081086828, 12.1499536241, 13.592286388, 14.1081230548, 15.2621223361, 15.4425168943, 16.2105519665, 16.3289342707, 17.4643953306, 17.518772004, 18.3883990965, 19.1444693222, 19.6933997517

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