Properties

Label 4-6720e2-1.1-c1e2-0-15
Degree $4$
Conductor $45158400$
Sign $1$
Analytic cond. $2879.33$
Root an. cond. $7.32526$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s − 2·5-s + 2·7-s + 3·9-s + 4·15-s − 4·21-s + 4·23-s + 3·25-s − 4·27-s − 8·29-s − 4·31-s − 4·35-s − 8·37-s + 4·41-s − 12·43-s − 6·45-s + 4·47-s + 3·49-s − 4·53-s + 8·59-s − 16·61-s + 6·63-s + 4·67-s − 8·69-s + 4·71-s + 8·73-s − 6·75-s + ⋯
L(s)  = 1  − 1.15·3-s − 0.894·5-s + 0.755·7-s + 9-s + 1.03·15-s − 0.872·21-s + 0.834·23-s + 3/5·25-s − 0.769·27-s − 1.48·29-s − 0.718·31-s − 0.676·35-s − 1.31·37-s + 0.624·41-s − 1.82·43-s − 0.894·45-s + 0.583·47-s + 3/7·49-s − 0.549·53-s + 1.04·59-s − 2.04·61-s + 0.755·63-s + 0.488·67-s − 0.963·69-s + 0.474·71-s + 0.936·73-s − 0.692·75-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 45158400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45158400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(45158400\)    =    \(2^{12} \cdot 3^{2} \cdot 5^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(2879.33\)
Root analytic conductor: \(7.32526\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 45158400,\ (\ :1/2, 1/2),\ 1)\)

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$C_1$ \( ( 1 + T )^{2} \)
5$C_1$ \( ( 1 + T )^{2} \)
7$C_1$ \( ( 1 - T )^{2} \)
good11$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 + p T^{2} )^{2} \)
23$D_{4}$ \( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
29$C_4$ \( 1 + 8 T + 54 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
31$C_4$ \( 1 + 4 T + 46 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 + 8 T + 70 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
41$C_4$ \( 1 - 4 T + 6 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 + 12 T + 102 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 - 4 T + 78 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
53$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
61$C_4$ \( 1 + 16 T + 166 T^{2} + 16 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 - 4 T + 118 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 - 4 T - 34 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 8 T + 142 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 - 8 T + 94 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - 16 T + 150 T^{2} - 16 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 - 4 T + 102 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
97$C_2^2$ \( 1 + 174 T^{2} + p^{2} T^{4} \)
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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

−7.58938889458375342114894915733, −7.58400705768765675929275648209, −7.04067241544700805494926835356, −6.86185048860045792638341218276, −6.32014825213729218481076959664, −6.19601871327697399192661268510, −5.40197774647185895400154717762, −5.38938401713916874867455528795, −4.90500382662263215337757360449, −4.85546647127553761811008252183, −4.11736709701848401100996505607, −3.92537317304507086089953496209, −3.43444876228649193080650819358, −3.22105032619658078629174650559, −2.25594340468474005481396056674, −2.12506417643249451370148878656, −1.23320469953243744914173468680, −1.16515863644149473107174368034, 0, 0, 1.16515863644149473107174368034, 1.23320469953243744914173468680, 2.12506417643249451370148878656, 2.25594340468474005481396056674, 3.22105032619658078629174650559, 3.43444876228649193080650819358, 3.92537317304507086089953496209, 4.11736709701848401100996505607, 4.85546647127553761811008252183, 4.90500382662263215337757360449, 5.38938401713916874867455528795, 5.40197774647185895400154717762, 6.19601871327697399192661268510, 6.32014825213729218481076959664, 6.86185048860045792638341218276, 7.04067241544700805494926835356, 7.58400705768765675929275648209, 7.58938889458375342114894915733

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