Properties

Label 4-672e2-1.1-c1e2-0-33
Degree $4$
Conductor $451584$
Sign $1$
Analytic cond. $28.7933$
Root an. cond. $2.31645$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 5·7-s − 2·11-s + 2·13-s + 2·17-s + 5·19-s − 5·21-s + 6·23-s + 5·25-s + 27-s − 16·29-s − 3·31-s + 2·33-s + 9·37-s − 2·39-s + 4·41-s − 2·43-s + 8·47-s + 18·49-s − 2·51-s − 6·53-s − 5·57-s + 6·59-s + 2·61-s − 5·67-s − 6·69-s − 8·71-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.88·7-s − 0.603·11-s + 0.554·13-s + 0.485·17-s + 1.14·19-s − 1.09·21-s + 1.25·23-s + 25-s + 0.192·27-s − 2.97·29-s − 0.538·31-s + 0.348·33-s + 1.47·37-s − 0.320·39-s + 0.624·41-s − 0.304·43-s + 1.16·47-s + 18/7·49-s − 0.280·51-s − 0.824·53-s − 0.662·57-s + 0.781·59-s + 0.256·61-s − 0.610·67-s − 0.722·69-s − 0.949·71-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 451584 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 451584 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(451584\)    =    \(2^{10} \cdot 3^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(28.7933\)
Root analytic conductor: \(2.31645\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 451584,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.106699033\)
\(L(\frac12)\) \(\approx\) \(2.106699033\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T + T^{2} \)
7$C_2$ \( 1 - 5 T + p T^{2} \)
good5$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 + 2 T - 7 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
17$C_2^2$ \( 1 - 2 T - 13 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
19$C_2^2$ \( 1 - 5 T + 6 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - 6 T + 13 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
31$C_2^2$ \( 1 + 3 T - 22 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
37$C_2^2$ \( 1 - 9 T + 44 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 - 8 T + 17 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 6 T - 17 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - 6 T - 23 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 2 T - 57 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
67$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
71$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 11 T + 48 T^{2} - 11 p T^{3} + p^{2} T^{4} \)
79$C_2^2$ \( 1 + 5 T - 54 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 + p T^{2} )^{2} \)
89$C_2^2$ \( 1 + 12 T + 55 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
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

−10.78021455451905773404204037605, −10.72178110063978566982942231493, −9.810955093394051389850436204406, −9.361206517325461333331800987441, −9.062847007752216879065560580631, −8.527646641628852897624022693000, −8.016209599977136680733833367634, −7.63642338093959683940699520044, −7.30801527687310916749354969058, −6.95791951708143720573156384382, −5.94593959996572713318420482631, −5.74154115195571844214046912106, −5.19913840645658210751736263716, −5.01552845589908719498520211227, −4.32511363872495015033309837038, −3.76925543571763609525423016555, −3.05826479198293593783990969297, −2.32570691715105480230014898929, −1.52305722575023184990589717115, −0.902854749428953579942752075037, 0.902854749428953579942752075037, 1.52305722575023184990589717115, 2.32570691715105480230014898929, 3.05826479198293593783990969297, 3.76925543571763609525423016555, 4.32511363872495015033309837038, 5.01552845589908719498520211227, 5.19913840645658210751736263716, 5.74154115195571844214046912106, 5.94593959996572713318420482631, 6.95791951708143720573156384382, 7.30801527687310916749354969058, 7.63642338093959683940699520044, 8.016209599977136680733833367634, 8.527646641628852897624022693000, 9.062847007752216879065560580631, 9.361206517325461333331800987441, 9.810955093394051389850436204406, 10.72178110063978566982942231493, 10.78021455451905773404204037605

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