Properties

Degree 4
Conductor $ 3^{2} \cdot 5^{2} \cdot 17^{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  + 2·3-s − 4·4-s + 3·5-s − 8·7-s + 3·9-s − 8·12-s + 6·15-s + 12·16-s − 17-s − 2·19-s − 12·20-s − 16·21-s + 18·23-s + 4·25-s + 4·27-s + 32·28-s − 24·35-s − 12·36-s − 8·37-s + 9·45-s + 24·48-s + 34·49-s − 2·51-s − 4·57-s + 12·59-s − 24·60-s − 24·63-s + ⋯
L(s)  = 1  + 1.15·3-s − 2·4-s + 1.34·5-s − 3.02·7-s + 9-s − 2.30·12-s + 1.54·15-s + 3·16-s − 0.242·17-s − 0.458·19-s − 2.68·20-s − 3.49·21-s + 3.75·23-s + 4/5·25-s + 0.769·27-s + 6.04·28-s − 4.05·35-s − 2·36-s − 1.31·37-s + 1.34·45-s + 3.46·48-s + 34/7·49-s − 0.280·51-s − 0.529·57-s + 1.56·59-s − 3.09·60-s − 3.02·63-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 1105425 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 1105425 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(1105425\)    =    \(3^{2} \cdot 5^{2} \cdot 17^{3}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{1105425} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 1105425,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $1.444174087$
$L(\frac12)$  $\approx$  $1.444174087$
$L(\frac{3}{2})$   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$C_1$ \( ( 1 - T )^{2} \)
5$C_2$ \( 1 - 3 T + p T^{2} \)
17$C_1$ \( 1 + T \)
good2$C_2$ \( ( 1 + p T^{2} )^{2} \)
7$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
11$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
13$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
19$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
31$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
37$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
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$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
71$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
73$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
89$C_2$ \( ( 1 + p T^{2} )^{2} \)
97$C_2$ \( ( 1 + 16 T + p T^{2} )^{2} \)
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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

−8.569700646198540476002149945968, −7.63075451299586448757437920341, −7.08199582321199217163758748774, −6.66463575826783700250676263362, −6.51609740369614917462879752898, −5.78200824128863401364705700099, −5.20289650874661325264421481027, −5.14306131647545403565477997005, −4.22172406081991876402843560872, −3.81905951450552627127186117098, −3.25257599736447137754746076902, −3.00632648437536992753290178800, −2.56940330683685764435785210432, −1.37557520782302937993720081395, −0.55449800498610258560594445237, 0.55449800498610258560594445237, 1.37557520782302937993720081395, 2.56940330683685764435785210432, 3.00632648437536992753290178800, 3.25257599736447137754746076902, 3.81905951450552627127186117098, 4.22172406081991876402843560872, 5.14306131647545403565477997005, 5.20289650874661325264421481027, 5.78200824128863401364705700099, 6.51609740369614917462879752898, 6.66463575826783700250676263362, 7.08199582321199217163758748774, 7.63075451299586448757437920341, 8.569700646198540476002149945968

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