Properties

Label 4-3120e2-1.1-c1e2-0-25
Degree $4$
Conductor $9734400$
Sign $1$
Analytic cond. $620.673$
Root an. cond. $4.99132$
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  + 4·5-s − 9-s + 12·11-s + 4·19-s + 11·25-s + 20·29-s − 8·31-s + 20·41-s − 4·45-s − 2·49-s + 48·55-s − 12·59-s − 12·61-s − 16·79-s + 81-s − 28·89-s + 16·95-s − 12·99-s + 20·101-s − 8·109-s + 86·121-s + 24·125-s + 127-s + 131-s + 137-s + 139-s + 80·145-s + ⋯
L(s)  = 1  + 1.78·5-s − 1/3·9-s + 3.61·11-s + 0.917·19-s + 11/5·25-s + 3.71·29-s − 1.43·31-s + 3.12·41-s − 0.596·45-s − 2/7·49-s + 6.47·55-s − 1.56·59-s − 1.53·61-s − 1.80·79-s + 1/9·81-s − 2.96·89-s + 1.64·95-s − 1.20·99-s + 1.99·101-s − 0.766·109-s + 7.81·121-s + 2.14·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 6.64·145-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9734400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9734400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(9734400\)    =    \(2^{8} \cdot 3^{2} \cdot 5^{2} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(620.673\)
Root analytic conductor: \(4.99132\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 9734400,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(6.856543366\)
\(L(\frac12)\) \(\approx\) \(6.856543366\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T^{2} \)
5$C_2$ \( 1 - 4 T + p T^{2} \)
13$C_2$ \( 1 + T^{2} \)
good7$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
17$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 - 38 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - p T^{2} )^{2} \)
47$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 142 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 150 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
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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

−8.958040616746122333111201474991, −8.663198519662771624716587161983, −8.424829985350713828616220005511, −7.66689760879511475984574926285, −7.09351890909329506586324070245, −7.05381595381731855820387239897, −6.42666367018735506774205082162, −6.21263921756240562714508946679, −5.96078293697032060894740412650, −5.76832631751107995023179159508, −4.92785139816646120842257796339, −4.65945338962121549324356948746, −4.22477599879419634678394927328, −3.85582733959903566998354501762, −3.05744024629474237944457919649, −2.96158915716007055894373938844, −2.27845855051693657560688095724, −1.57864541294792856600228268624, −1.18695580078384063840761271952, −1.01865613135659887529208141765, 1.01865613135659887529208141765, 1.18695580078384063840761271952, 1.57864541294792856600228268624, 2.27845855051693657560688095724, 2.96158915716007055894373938844, 3.05744024629474237944457919649, 3.85582733959903566998354501762, 4.22477599879419634678394927328, 4.65945338962121549324356948746, 4.92785139816646120842257796339, 5.76832631751107995023179159508, 5.96078293697032060894740412650, 6.21263921756240562714508946679, 6.42666367018735506774205082162, 7.05381595381731855820387239897, 7.09351890909329506586324070245, 7.66689760879511475984574926285, 8.424829985350713828616220005511, 8.663198519662771624716587161983, 8.958040616746122333111201474991

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