Properties

Label 4-2880e2-1.1-c1e2-0-2
Degree $4$
Conductor $8294400$
Sign $1$
Analytic cond. $528.858$
Root an. cond. $4.79550$
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 − 8·11-s + 8·19-s + 11·25-s − 8·29-s + 16·41-s − 2·49-s + 32·55-s − 24·59-s − 4·61-s − 16·71-s − 16·79-s − 32·95-s + 24·101-s + 36·109-s + 26·121-s − 24·125-s + 127-s + 131-s + 137-s + 139-s + 32·145-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯
L(s)  = 1  − 1.78·5-s − 2.41·11-s + 1.83·19-s + 11/5·25-s − 1.48·29-s + 2.49·41-s − 2/7·49-s + 4.31·55-s − 3.12·59-s − 0.512·61-s − 1.89·71-s − 1.80·79-s − 3.28·95-s + 2.38·101-s + 3.44·109-s + 2.36·121-s − 2.14·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 2.65·145-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8294400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8294400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(8294400\)    =    \(2^{12} \cdot 3^{4} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(528.858\)
Root analytic conductor: \(4.79550\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{2880} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 8294400,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4281290688\)
\(L(\frac12)\) \(\approx\) \(0.4281290688\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + 4 T + p T^{2} \)
good7$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
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$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + p T^{2} )^{2} \)
37$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - p T^{2} )^{2} \)
47$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 102 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
79$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 102 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \)
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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

−9.024194744153206254052796956394, −8.487461283048842591902316166351, −7.978546430325502216157461695627, −7.68960445231581864247947747936, −7.62367486874556955243547348235, −7.24312746076055942709068405075, −7.11283471984395898066137506535, −6.00054360703214209432234930621, −5.96378065035739271085672422017, −5.54583389924879965641187452760, −4.91872394632157759872463135633, −4.59963128567168788112929423962, −4.48302779728625038791681746606, −3.57508696283931694946623964982, −3.43762370323646555213323837738, −2.82850980720148956160380265430, −2.69508607498131347000657334982, −1.81178749622729775066131397465, −1.01009861874939604786914291011, −0.24569304183740077259754034444, 0.24569304183740077259754034444, 1.01009861874939604786914291011, 1.81178749622729775066131397465, 2.69508607498131347000657334982, 2.82850980720148956160380265430, 3.43762370323646555213323837738, 3.57508696283931694946623964982, 4.48302779728625038791681746606, 4.59963128567168788112929423962, 4.91872394632157759872463135633, 5.54583389924879965641187452760, 5.96378065035739271085672422017, 6.00054360703214209432234930621, 7.11283471984395898066137506535, 7.24312746076055942709068405075, 7.62367486874556955243547348235, 7.68960445231581864247947747936, 7.978546430325502216157461695627, 8.487461283048842591902316166351, 9.024194744153206254052796956394

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