Properties

Degree 4
Conductor $ 2^{2} \cdot 17^{2} \cdot 53^{2} $
Sign $-1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 1

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 4-s − 8·7-s − 2·9-s + 12·11-s + 4·13-s + 16-s − 2·17-s − 10·25-s − 8·28-s − 2·36-s − 8·37-s + 16·43-s + 12·44-s + 34·49-s + 4·52-s − 6·53-s + 16·63-s + 64-s − 2·68-s − 96·77-s − 5·81-s − 12·89-s − 32·91-s + 28·97-s − 24·99-s − 10·100-s − 12·107-s + ⋯
L(s)  = 1  + 1/2·4-s − 3.02·7-s − 2/3·9-s + 3.61·11-s + 1.10·13-s + 1/4·16-s − 0.485·17-s − 2·25-s − 1.51·28-s − 1/3·36-s − 1.31·37-s + 2.43·43-s + 1.80·44-s + 34/7·49-s + 0.554·52-s − 0.824·53-s + 2.01·63-s + 1/8·64-s − 0.242·68-s − 10.9·77-s − 5/9·81-s − 1.27·89-s − 3.35·91-s + 2.84·97-s − 2.41·99-s − 100-s − 1.16·107-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 3247204 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 3247204 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(3247204\)    =    \(2^{2} \cdot 17^{2} \cdot 53^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{3247204} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  no
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(4,\ 3247204,\ (\ :1/2, 1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

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

−7.13690330956485355862455539993, −6.65330731261113455462227374582, −6.51750679489306161330527852278, −6.07940975059845120120491473895, −5.99911855171913780844071238816, −5.64127450416579378652046822039, −4.49730750500575581109364219588, −3.90229547122613769308947697962, −3.80320802252311147126867436038, −3.49677870190616727260642617802, −3.01941537763985245794009266319, −2.33242998394544666678362686278, −1.60618933200325615024936212313, −0.961208165563165598163043574434, 0, 0.961208165563165598163043574434, 1.60618933200325615024936212313, 2.33242998394544666678362686278, 3.01941537763985245794009266319, 3.49677870190616727260642617802, 3.80320802252311147126867436038, 3.90229547122613769308947697962, 4.49730750500575581109364219588, 5.64127450416579378652046822039, 5.99911855171913780844071238816, 6.07940975059845120120491473895, 6.51750679489306161330527852278, 6.65330731261113455462227374582, 7.13690330956485355862455539993

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