Properties

Label 4-720e2-1.1-c1e2-0-9
Degree $4$
Conductor $518400$
Sign $1$
Analytic cond. $33.0536$
Root an. cond. $2.39775$
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  + 2·5-s − 8·11-s − 8·19-s − 25-s + 4·29-s − 4·41-s + 10·49-s − 16·55-s + 24·59-s − 20·61-s + 16·71-s + 32·79-s + 12·89-s − 16·95-s − 12·101-s + 12·109-s + 26·121-s − 12·125-s + 127-s + 131-s + 137-s + 139-s + 8·145-s + 149-s + 151-s + 157-s + 163-s + ⋯
L(s)  = 1  + 0.894·5-s − 2.41·11-s − 1.83·19-s − 1/5·25-s + 0.742·29-s − 0.624·41-s + 10/7·49-s − 2.15·55-s + 3.12·59-s − 2.56·61-s + 1.89·71-s + 3.60·79-s + 1.27·89-s − 1.64·95-s − 1.19·101-s + 1.14·109-s + 2.36·121-s − 1.07·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.664·145-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 518400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 518400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(518400\)    =    \(2^{8} \cdot 3^{4} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(33.0536\)
Root analytic conductor: \(2.39775\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 518400,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.429543584\)
\(L(\frac12)\) \(\approx\) \(1.429543584\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5$C_2$ \( 1 - 2 T + p T^{2} \)
good7$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
17$C_2$ \( ( 1 - p T^{2} )^{2} \)
19$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - 42 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + p T^{2} )^{2} \)
37$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
59$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 + 62 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 82 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 16 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 162 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 + 62 T^{2} + p^{2} T^{4} \)
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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

−10.69624274799863726152397370424, −10.41957516066490583740359680763, −9.895211204366680163356715472970, −9.300063373497798212663900221466, −9.022982550101318757189098588511, −8.276342137901003229483989860351, −8.133884297769304140981235323564, −7.76827312540848623439299005793, −7.08880796671355039986654818950, −6.55522606158342277188410955736, −6.27952834674104816965489366041, −5.45469305988578750453821885930, −5.44294100118225694791529000095, −4.82460444005880207585720832866, −4.27430069994080419571574218214, −3.57907100035208770629193334516, −2.83190990143063948492180412278, −2.18795827357141964863149142728, −2.09572296176541280754924722822, −0.59036344855411467976633806951, 0.59036344855411467976633806951, 2.09572296176541280754924722822, 2.18795827357141964863149142728, 2.83190990143063948492180412278, 3.57907100035208770629193334516, 4.27430069994080419571574218214, 4.82460444005880207585720832866, 5.44294100118225694791529000095, 5.45469305988578750453821885930, 6.27952834674104816965489366041, 6.55522606158342277188410955736, 7.08880796671355039986654818950, 7.76827312540848623439299005793, 8.133884297769304140981235323564, 8.276342137901003229483989860351, 9.022982550101318757189098588511, 9.300063373497798212663900221466, 9.895211204366680163356715472970, 10.41957516066490583740359680763, 10.69624274799863726152397370424

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