Properties

Degree 4
Conductor $ 3^{3} \cdot 7^{2} \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 + 6·5-s − 4·7-s + 9-s − 4·12-s + 6·15-s + 12·16-s − 2·17-s − 24·20-s − 4·21-s + 17·25-s + 27-s + 16·28-s − 24·35-s − 4·36-s − 8·37-s − 6·41-s − 14·43-s + 6·45-s − 12·47-s + 12·48-s + 9·49-s − 2·51-s + 12·59-s − 24·60-s − 4·63-s + ⋯
L(s)  = 1  + 0.577·3-s − 2·4-s + 2.68·5-s − 1.51·7-s + 1/3·9-s − 1.15·12-s + 1.54·15-s + 3·16-s − 0.485·17-s − 5.36·20-s − 0.872·21-s + 17/5·25-s + 0.192·27-s + 3.02·28-s − 4.05·35-s − 2/3·36-s − 1.31·37-s − 0.937·41-s − 2.13·43-s + 0.894·45-s − 1.75·47-s + 1.73·48-s + 9/7·49-s − 0.280·51-s + 1.56·59-s − 3.09·60-s − 0.503·63-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 382347 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 382347 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(382347\)    =    \(3^{3} \cdot 7^{2} \cdot 17^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{382347} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  no
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(4,\ 382347,\ (\ :1/2, 1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;7,\;17\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{3,\;7,\;17\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad3$C_1$ \( 1 - T \)
7$C_2$ \( 1 + 4 T + p T^{2} \)
17$C_1$ \( ( 1 + T )^{2} \)
good2$C_2$ \( ( 1 + p T^{2} )^{2} \)
5$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
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$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
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$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
37$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
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$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
73$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
79$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
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

−8.618966621189813626434174913816, −8.469847884947395544150995937007, −7.58523751895654898654671253270, −6.71025591249190433133409575281, −6.66463575826783700250676263362, −5.91938342687192026845077841351, −5.62865745280264567981666866924, −5.14306131647545403565477997005, −4.71647271720364633813685518770, −3.96826933720343816290629867972, −3.27969494379582339775711456114, −3.00632648437536992753290178800, −1.98195814340014591526633802801, −1.44650813195969740431651518332, 0, 1.44650813195969740431651518332, 1.98195814340014591526633802801, 3.00632648437536992753290178800, 3.27969494379582339775711456114, 3.96826933720343816290629867972, 4.71647271720364633813685518770, 5.14306131647545403565477997005, 5.62865745280264567981666866924, 5.91938342687192026845077841351, 6.66463575826783700250676263362, 6.71025591249190433133409575281, 7.58523751895654898654671253270, 8.469847884947395544150995937007, 8.618966621189813626434174913816

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