Properties

Degree 4
Conductor $ 2^{2} \cdot 5^{2} \cdot 13^{3} $
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  + 2·2-s + 3·4-s − 5-s + 2·7-s + 4·8-s + 3·9-s − 2·10-s − 13-s + 4·14-s + 5·16-s + 6·18-s − 3·20-s − 4·25-s − 2·26-s + 6·28-s + 4·29-s + 6·32-s − 2·35-s + 9·36-s + 6·37-s − 4·40-s − 3·45-s + 26·47-s − 11·49-s − 8·50-s − 3·52-s + 8·56-s + ⋯
L(s)  = 1  + 1.41·2-s + 3/2·4-s − 0.447·5-s + 0.755·7-s + 1.41·8-s + 9-s − 0.632·10-s − 0.277·13-s + 1.06·14-s + 5/4·16-s + 1.41·18-s − 0.670·20-s − 4/5·25-s − 0.392·26-s + 1.13·28-s + 0.742·29-s + 1.06·32-s − 0.338·35-s + 3/2·36-s + 0.986·37-s − 0.632·40-s − 0.447·45-s + 3.79·47-s − 1.57·49-s − 1.13·50-s − 0.416·52-s + 1.06·56-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 219700 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 219700 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(219700\)    =    \(2^{2} \cdot 5^{2} \cdot 13^{3}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{219700} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 219700,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $4.687099046$
$L(\frac12)$  $\approx$  $4.687099046$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

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

−9.026911386413610630551375476230, −8.420872023290300609672133626170, −7.78618160341898961599148312743, −7.45985560667588965978165906144, −7.23568791400333437059684221061, −6.51632105010820945514209338703, −5.88908450915949561478136129218, −5.69499349255850374306805786509, −4.65916643268043458244706899373, −4.61153453491182141362175201622, −4.15748343893950010861474305754, −3.45971551695615743734182930904, −2.76910930427420910776332481273, −2.04502161572044176240278653055, −1.23311319217689188134168770571, 1.23311319217689188134168770571, 2.04502161572044176240278653055, 2.76910930427420910776332481273, 3.45971551695615743734182930904, 4.15748343893950010861474305754, 4.61153453491182141362175201622, 4.65916643268043458244706899373, 5.69499349255850374306805786509, 5.88908450915949561478136129218, 6.51632105010820945514209338703, 7.23568791400333437059684221061, 7.45985560667588965978165906144, 7.78618160341898961599148312743, 8.420872023290300609672133626170, 9.026911386413610630551375476230

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