Properties

Degree 4
Conductor $ 3^{3} \cdot 17^{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  + 3-s − 4·4-s − 8·7-s + 9-s − 4·12-s − 2·13-s + 12·16-s − 2·19-s − 8·21-s − 25-s + 27-s + 32·28-s + 4·31-s − 4·36-s − 8·37-s − 2·39-s − 14·43-s + 12·48-s + 34·49-s + 8·52-s − 2·57-s + 16·61-s − 8·63-s − 32·64-s − 8·67-s + 4·73-s − 75-s + ⋯
L(s)  = 1  + 0.577·3-s − 2·4-s − 3.02·7-s + 1/3·9-s − 1.15·12-s − 0.554·13-s + 3·16-s − 0.458·19-s − 1.74·21-s − 1/5·25-s + 0.192·27-s + 6.04·28-s + 0.718·31-s − 2/3·36-s − 1.31·37-s − 0.320·39-s − 2.13·43-s + 1.73·48-s + 34/7·49-s + 1.10·52-s − 0.264·57-s + 2.04·61-s − 1.00·63-s − 4·64-s − 0.977·67-s + 0.468·73-s − 0.115·75-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 7803 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 7803 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(7803\)    =    \(3^{3} \cdot 17^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{7803} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  no
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(4,\ 7803,\ (\ :1/2, 1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;17\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{3,\;17\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad3$C_1$ \( 1 - T \)
17$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good2$C_2$ \( ( 1 + p T^{2} )^{2} \)
5$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
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$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
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$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
31$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
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$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
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$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
61$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
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$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
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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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−11.81013235135078842473393257576, −10.23827402727964604848387184191, −10.23122640158841960226025987177, −9.642727340101070443548067847258, −9.392750473138958647961697264371, −8.618966621189813626434174913816, −8.382258919016960888464931958518, −7.25132554126425354900557687617, −6.66463575826783700250676263362, −5.94155321226588727412442600100, −5.14306131647545403565477997005, −4.19114262883338708198809215469, −3.52343415212056409865262439727, −3.00632648437536992753290178800, 0, 3.00632648437536992753290178800, 3.52343415212056409865262439727, 4.19114262883338708198809215469, 5.14306131647545403565477997005, 5.94155321226588727412442600100, 6.66463575826783700250676263362, 7.25132554126425354900557687617, 8.382258919016960888464931958518, 8.618966621189813626434174913816, 9.392750473138958647961697264371, 9.642727340101070443548067847258, 10.23122640158841960226025987177, 10.23827402727964604848387184191, 11.81013235135078842473393257576

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