Properties

Label 4-114075-1.1-c1e2-0-20
Degree $4$
Conductor $114075$
Sign $-1$
Analytic cond. $7.27352$
Root an. cond. $1.64223$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $1$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 2·7-s + 9-s − 2·13-s − 4·16-s − 12·19-s − 2·21-s + 25-s + 27-s + 4·31-s + 14·37-s − 2·39-s − 16·43-s − 4·48-s − 11·49-s − 12·57-s + 10·61-s − 2·63-s − 8·67-s − 12·73-s + 75-s − 6·79-s + 81-s + 4·91-s + 4·93-s − 22·97-s − 8·103-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.755·7-s + 1/3·9-s − 0.554·13-s − 16-s − 2.75·19-s − 0.436·21-s + 1/5·25-s + 0.192·27-s + 0.718·31-s + 2.30·37-s − 0.320·39-s − 2.43·43-s − 0.577·48-s − 1.57·49-s − 1.58·57-s + 1.28·61-s − 0.251·63-s − 0.977·67-s − 1.40·73-s + 0.115·75-s − 0.675·79-s + 1/9·81-s + 0.419·91-s + 0.414·93-s − 2.23·97-s − 0.788·103-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: $-1$
Analytic conductor: \(7.27352\)
Root analytic conductor: \(1.64223\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
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$C_2$ \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \)
7$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
11$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
19$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
29$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
31$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \)
43$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \)
73$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \)
97$C_2$ \( ( 1 + 11 T + p T^{2} )^{2} \)
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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

−9.348864711978766678634703610950, −8.599163171775301233119270744200, −8.333107995317450905784282789603, −7.997850874518894594264893149223, −6.93279974611194334514178729187, −6.92169485416648038433791572415, −6.28215256608024799923334344773, −5.86662536743360060967332495720, −4.75380031772719504345197730313, −4.52648172003467775210616179348, −3.93328920262448493706980455054, −3.02709857296590648239002053477, −2.53020079052262607160250914109, −1.78483850539275904424541360817, 0, 1.78483850539275904424541360817, 2.53020079052262607160250914109, 3.02709857296590648239002053477, 3.93328920262448493706980455054, 4.52648172003467775210616179348, 4.75380031772719504345197730313, 5.86662536743360060967332495720, 6.28215256608024799923334344773, 6.92169485416648038433791572415, 6.93279974611194334514178729187, 7.997850874518894594264893149223, 8.333107995317450905784282789603, 8.599163171775301233119270744200, 9.348864711978766678634703610950

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