Properties

Label 4-6720e2-1.1-c1e2-0-7
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 $0$

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 − 2·11-s − 2·13-s − 4·15-s + 4·17-s − 6·19-s − 4·21-s + 3·25-s − 4·27-s + 4·29-s + 2·31-s + 4·33-s + 4·35-s + 4·37-s + 4·39-s + 4·41-s − 4·43-s + 6·45-s + 3·49-s − 8·51-s − 2·53-s − 4·55-s + 12·57-s + 8·59-s + ⋯
L(s)  = 1  − 1.15·3-s + 0.894·5-s + 0.755·7-s + 9-s − 0.603·11-s − 0.554·13-s − 1.03·15-s + 0.970·17-s − 1.37·19-s − 0.872·21-s + 3/5·25-s − 0.769·27-s + 0.742·29-s + 0.359·31-s + 0.696·33-s + 0.676·35-s + 0.657·37-s + 0.640·39-s + 0.624·41-s − 0.609·43-s + 0.894·45-s + 3/7·49-s − 1.12·51-s − 0.274·53-s − 0.539·55-s + 1.58·57-s + 1.04·59-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: induced by $\chi_{6720} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 45158400,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.932925725\)
\(L(\frac12)\) \(\approx\) \(2.932925725\)
\(L(\frac{3}{2})\) 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$D_{4}$ \( 1 + 2 T + 6 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
13$C_4$ \( 1 + 2 T + 10 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
17$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
19$D_{4}$ \( 1 + 6 T + 30 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
31$D_{4}$ \( 1 - 2 T + 46 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
43$D_{4}$ \( 1 + 4 T + 22 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$D_{4}$ \( 1 + 2 T - 46 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
59$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
61$C_2^2$ \( 1 + 54 T^{2} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + 12 T + 102 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + 2 T + 126 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 22 T + 250 T^{2} - 22 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 - 12 T + 126 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
89$D_{4}$ \( 1 - 16 T + 174 T^{2} - 16 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 18 T + 258 T^{2} - 18 p T^{3} + p^{2} T^{4} \)
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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

−8.142742959714689961919246396233, −7.82389758625058590301950783520, −7.26319952538656243072310933473, −7.25164144335442151100501031169, −6.46946987729039197030577977881, −6.45383063273886412075943074750, −5.96153218349929408049912422304, −5.83187378769775070657510700050, −5.14732897093301345743613240073, −5.09880520705935399400457081385, −4.67305829474337271056714611224, −4.51102712715630081017310728516, −3.77147443368869694533950532494, −3.54183394192513226373643460769, −2.71480837017893321977452870093, −2.52273585610845977470953891964, −1.84208102599593655266593948468, −1.73919828323421313220491907440, −0.71008858931348487960499533584, −0.68727139888052538684776912316, 0.68727139888052538684776912316, 0.71008858931348487960499533584, 1.73919828323421313220491907440, 1.84208102599593655266593948468, 2.52273585610845977470953891964, 2.71480837017893321977452870093, 3.54183394192513226373643460769, 3.77147443368869694533950532494, 4.51102712715630081017310728516, 4.67305829474337271056714611224, 5.09880520705935399400457081385, 5.14732897093301345743613240073, 5.83187378769775070657510700050, 5.96153218349929408049912422304, 6.45383063273886412075943074750, 6.46946987729039197030577977881, 7.25164144335442151100501031169, 7.26319952538656243072310933473, 7.82389758625058590301950783520, 8.142742959714689961919246396233

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