Properties

Degree 4
Conductor $ 2^{7} \cdot 3^{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-s − 3·3-s + 4-s − 2·5-s − 3·6-s + 8-s + 6·9-s − 2·10-s − 3·12-s + 6·15-s + 16-s + 6·18-s + 12·19-s − 2·20-s − 8·23-s − 3·24-s − 7·25-s − 9·27-s + 4·29-s + 6·30-s + 32-s + 6·36-s + 12·38-s − 2·40-s − 10·43-s − 12·45-s − 8·46-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.73·3-s + 1/2·4-s − 0.894·5-s − 1.22·6-s + 0.353·8-s + 2·9-s − 0.632·10-s − 0.866·12-s + 1.54·15-s + 1/4·16-s + 1.41·18-s + 2.75·19-s − 0.447·20-s − 1.66·23-s − 0.612·24-s − 7/5·25-s − 1.73·27-s + 0.742·29-s + 1.09·30-s + 0.176·32-s + 36-s + 1.94·38-s − 0.316·40-s − 1.52·43-s − 1.78·45-s − 1.17·46-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 194688 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 194688 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(194688\)    =    \(2^{7} \cdot 3^{2} \cdot 13^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{194688} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 194688,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $1.101937971$
$L(\frac12)$  $\approx$  $1.101937971$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;13\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;3,\;13\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad2$C_1$ \( 1 - T \)
3$C_2$ \( 1 + p T + p T^{2} \)
13$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good5$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
7$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
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$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
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$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
41$C_2$ \( ( 1 + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 13 T + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
61$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
67$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
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

−9.157012014457056949579746961332, −8.598048288221075604337638832752, −7.75736614926909516215671940400, −7.45985560667588965978165906144, −7.30327241160797448656855232337, −6.54569148644882688306698852608, −5.88908450915949561478136129218, −5.72354157710313112451215291472, −5.23968918694001494220241942346, −4.61153453491182141362175201622, −4.03158163467550651987640538192, −3.70601718958614569024875598396, −2.82349792109875632158224534151, −1.70482056505424981206732382452, −0.68551053241257569931593057960, 0.68551053241257569931593057960, 1.70482056505424981206732382452, 2.82349792109875632158224534151, 3.70601718958614569024875598396, 4.03158163467550651987640538192, 4.61153453491182141362175201622, 5.23968918694001494220241942346, 5.72354157710313112451215291472, 5.88908450915949561478136129218, 6.54569148644882688306698852608, 7.30327241160797448656855232337, 7.45985560667588965978165906144, 7.75736614926909516215671940400, 8.598048288221075604337638832752, 9.157012014457056949579746961332

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