Properties

Degree $4$
Conductor $97344$
Sign $-1$
Motivic weight $1$
Primitive no
Self-dual yes
Analytic rank $1$

Origins

Origins of factors

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s + 3·9-s − 2·13-s + 4·17-s − 16·23-s − 6·25-s − 4·27-s + 12·29-s + 4·39-s + 8·43-s − 14·49-s − 8·51-s − 4·53-s − 4·61-s + 32·69-s + 12·75-s − 16·79-s + 5·81-s − 24·87-s − 36·101-s + 32·103-s − 24·107-s + 36·113-s − 6·117-s − 6·121-s + 127-s − 16·129-s + ⋯
L(s)  = 1  − 1.15·3-s + 9-s − 0.554·13-s + 0.970·17-s − 3.33·23-s − 6/5·25-s − 0.769·27-s + 2.22·29-s + 0.640·39-s + 1.21·43-s − 2·49-s − 1.12·51-s − 0.549·53-s − 0.512·61-s + 3.85·69-s + 1.38·75-s − 1.80·79-s + 5/9·81-s − 2.57·87-s − 3.58·101-s + 3.15·103-s − 2.32·107-s + 3.38·113-s − 0.554·117-s − 0.545·121-s + 0.0887·127-s − 1.40·129-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 97344 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 97344 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(97344\)    =    \(2^{6} \cdot 3^{2} \cdot 13^{2}\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{97344} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 97344,\ (\ :1/2, 1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad2 \( 1 \)
3$C_1$ \( ( 1 + T )^{2} \)
13$C_2$ \( 1 + 2 T + p T^{2} \)
good5$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
7$C_2$ \( ( 1 + p T^{2} )^{2} \)
11$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
23$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
61$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
71$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
73$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
79$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
97$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
show more
show less
   \(L(s) = \displaystyle\prod_{\mathfrak{p}\ \mathrm{bad}} (1- a(\mathfrak{p}) (N\mathfrak{p})^{-s})^{-1} \prod_{\mathfrak{p}\ \mathrm{good}} (1- a(\mathfrak{p}) (N\mathfrak{p})^{-s} + (N\mathfrak{p})^{-2s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.723104635417270733664366812662, −8.868697600985378807868163739755, −8.098990694093691505068092710868, −7.87110425385464535734394581728, −7.45593059858380139400376597909, −6.42897107072744719896577758454, −6.40700571362089881432056934326, −5.71800144135285822069832219153, −5.34499067736636279987817630194, −4.42580395477182623401042922608, −4.25303028692796488061253338187, −3.34342864873719270741111415825, −2.36285920530122244583655914110, −1.45535556086435766842287970552, 0, 1.45535556086435766842287970552, 2.36285920530122244583655914110, 3.34342864873719270741111415825, 4.25303028692796488061253338187, 4.42580395477182623401042922608, 5.34499067736636279987817630194, 5.71800144135285822069832219153, 6.40700571362089881432056934326, 6.42897107072744719896577758454, 7.45593059858380139400376597909, 7.87110425385464535734394581728, 8.098990694093691505068092710868, 8.868697600985378807868163739755, 9.723104635417270733664366812662

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