Properties

Degree 4
Conductor $ 3^{2} \cdot 5^{2} \cdot 17^{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  − 4·4-s + 3·5-s + 9-s − 6·11-s + 12·16-s − 2·19-s − 12·20-s + 4·25-s + 12·29-s + 4·31-s − 4·36-s − 6·41-s + 24·44-s + 3·45-s + 2·49-s − 18·55-s + 12·59-s + 16·61-s − 32·64-s + 24·71-s + 8·76-s − 20·79-s + 36·80-s + 81-s − 6·95-s − 6·99-s − 16·100-s + ⋯
L(s)  = 1  − 2·4-s + 1.34·5-s + 1/3·9-s − 1.80·11-s + 3·16-s − 0.458·19-s − 2.68·20-s + 4/5·25-s + 2.22·29-s + 0.718·31-s − 2/3·36-s − 0.937·41-s + 3.61·44-s + 0.447·45-s + 2/7·49-s − 2.42·55-s + 1.56·59-s + 2.04·61-s − 4·64-s + 2.84·71-s + 0.917·76-s − 2.25·79-s + 4.02·80-s + 1/9·81-s − 0.615·95-s − 0.603·99-s − 8/5·100-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 65025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 65025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(65025\)    =    \(3^{2} \cdot 5^{2} \cdot 17^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{65025} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 65025,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $0.9924137172$
$L(\frac12)$  $\approx$  $0.9924137172$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;5,\;17\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{3,\;5,\;17\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad3$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
5$C_2$ \( 1 - 3 T + p T^{2} \)
17$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good2$C_2$ \( ( 1 + p T^{2} )^{2} \)
7$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
11$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
19$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
41$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
47$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
71$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
79$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
89$C_2$ \( ( 1 + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
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\[\begin{aligned} L(s) = \prod_{\mathfrak{p}\ \mathrm{bad}} (1- a(\mathfrak{p}) (N\mathfrak{p})^{-s})^{-1} \prod_{\mathfrak{p}\ \mathrm{good}} (1- a(\mathfrak{p}) (N\mathfrak{p})^{-s} + (N\mathfrak{p})^{-2s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−9.886646646989496430590747561470, −9.642727340101070443548067847258, −8.619064027245224362683221960799, −8.618966621189813626434174913816, −8.163914181664516579409971266285, −7.47209372332904761130246414384, −6.66463575826783700250676263362, −6.06854626571026470323908953719, −5.34999295252576679459237419348, −5.14306131647545403565477997005, −4.63447764212713603277414423316, −3.91640635015127100849092687113, −3.00632648437536992753290178800, −2.23913687121337646524702105205, −0.844124651194318703334441921678, 0.844124651194318703334441921678, 2.23913687121337646524702105205, 3.00632648437536992753290178800, 3.91640635015127100849092687113, 4.63447764212713603277414423316, 5.14306131647545403565477997005, 5.34999295252576679459237419348, 6.06854626571026470323908953719, 6.66463575826783700250676263362, 7.47209372332904761130246414384, 8.163914181664516579409971266285, 8.618966621189813626434174913816, 8.619064027245224362683221960799, 9.642727340101070443548067847258, 9.886646646989496430590747561470

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