Properties

Degree 4
Conductor $ 2^{2} \cdot 3^{2} \cdot 31^{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  + 4-s − 3·9-s + 4·13-s + 16-s + 8·19-s − 6·25-s − 2·31-s − 3·36-s + 20·37-s + 16·43-s − 14·49-s + 4·52-s − 12·61-s + 64-s − 24·67-s + 20·73-s + 8·76-s − 16·79-s + 9·81-s + 4·97-s − 6·100-s + 16·103-s − 4·109-s − 12·117-s − 22·121-s − 2·124-s + 127-s + ⋯
L(s)  = 1  + 1/2·4-s − 9-s + 1.10·13-s + 1/4·16-s + 1.83·19-s − 6/5·25-s − 0.359·31-s − 1/2·36-s + 3.28·37-s + 2.43·43-s − 2·49-s + 0.554·52-s − 1.53·61-s + 1/8·64-s − 2.93·67-s + 2.34·73-s + 0.917·76-s − 1.80·79-s + 81-s + 0.406·97-s − 3/5·100-s + 1.57·103-s − 0.383·109-s − 1.10·117-s − 2·121-s − 0.179·124-s + 0.0887·127-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 34596 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 34596 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(34596\)    =    \(2^{2} \cdot 3^{2} \cdot 31^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{34596} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 34596,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $1.433324264$
$L(\frac12)$  $\approx$  $1.433324264$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;31\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;3,\;31\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad2$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
3$C_2$ \( 1 + p T^{2} \)
31$C_1$ \( ( 1 + T )^{2} \)
good5$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
7$C_2$ \( ( 1 + p T^{2} )^{2} \)
11$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
29$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
43$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
61$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
73$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
97$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
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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

−10.50978028730293557207764308625, −9.792819761543788827804434951174, −9.258085561472781634099322215629, −9.056498267458590117851162621111, −8.059614510201906936269350354648, −7.76857292337562945056886541715, −7.42375835124516750954422223560, −6.32858579420662158347624019611, −6.01709210768183680394437750392, −5.66012783592677814232561850819, −4.74203912193695697715499891246, −3.93624561426330149363253797109, −3.16188554299090604324575557253, −2.56538619972077555026069053800, −1.25027517360414188495745145887, 1.25027517360414188495745145887, 2.56538619972077555026069053800, 3.16188554299090604324575557253, 3.93624561426330149363253797109, 4.74203912193695697715499891246, 5.66012783592677814232561850819, 6.01709210768183680394437750392, 6.32858579420662158347624019611, 7.42375835124516750954422223560, 7.76857292337562945056886541715, 8.059614510201906936269350354648, 9.056498267458590117851162621111, 9.258085561472781634099322215629, 9.792819761543788827804434951174, 10.50978028730293557207764308625

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