Properties

Degree $4$
Conductor $114075$
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 − 3·4-s + 9-s − 3·12-s + 2·13-s + 5·16-s − 8·19-s + 25-s + 27-s − 16·31-s − 3·36-s + 12·37-s + 2·39-s − 8·43-s + 5·48-s − 14·49-s − 6·52-s − 8·57-s − 4·61-s − 3·64-s − 8·67-s − 12·73-s + 75-s + 24·76-s + 32·79-s + 81-s − 16·93-s + ⋯
L(s)  = 1  + 0.577·3-s − 3/2·4-s + 1/3·9-s − 0.866·12-s + 0.554·13-s + 5/4·16-s − 1.83·19-s + 1/5·25-s + 0.192·27-s − 2.87·31-s − 1/2·36-s + 1.97·37-s + 0.320·39-s − 1.21·43-s + 0.721·48-s − 2·49-s − 0.832·52-s − 1.05·57-s − 0.512·61-s − 3/8·64-s − 0.977·67-s − 1.40·73-s + 0.115·75-s + 2.75·76-s + 3.60·79-s + 1/9·81-s − 1.65·93-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 114075 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 114075 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(114075\)    =    \(3^{3} \cdot 5^{2} \cdot 13^{2}\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{114075} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 114075,\ (\ :1/2, 1/2),\ -1)\)

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)$
bad3$C_1$ \( 1 - T \)
5$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
13$C_1$ \( ( 1 - T )^{2} \)
good2$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
7$C_2$ \( ( 1 + p T^{2} )^{2} \)
11$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
19$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
29$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
31$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
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$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
61$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 16 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
97$C_2$ \( ( 1 - 18 T + p T^{2} )^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.244674050330195584769830754850, −8.846234947699956116857977484454, −8.404296066573383401625737694753, −7.73233535694260719129401432761, −7.66093767427808108694408525270, −6.48654966834348628333399350819, −6.37803160504271454727036453417, −5.52200307165618101356281230656, −4.88823595164803979266926983864, −4.51284738641056261325871513649, −3.73104630189750446831233920392, −3.56799192480000330057521215275, −2.43381471255309129843962850009, −1.53827946418072491307494025034, 0, 1.53827946418072491307494025034, 2.43381471255309129843962850009, 3.56799192480000330057521215275, 3.73104630189750446831233920392, 4.51284738641056261325871513649, 4.88823595164803979266926983864, 5.52200307165618101356281230656, 6.37803160504271454727036453417, 6.48654966834348628333399350819, 7.66093767427808108694408525270, 7.73233535694260719129401432761, 8.404296066573383401625737694753, 8.846234947699956116857977484454, 9.244674050330195584769830754850

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