Properties

Degree $4$
Conductor $518400$
Sign $1$
Motivic weight $1$
Primitive no
Self-dual yes
Analytic rank $2$

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

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

Invariants

Degree: \(4\)
Conductor: \(518400\)    =    \(2^{8} \cdot 3^{4} \cdot 5^{2}\)
Sign: $1$
Motivic weight: \(1\)
Character: induced by $\chi_{720} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 518400,\ (\ :1/2, 1/2),\ 1)\)

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 \( 1 \)
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

−10.15868904826550110521612182023, −10.14648392519205838162628278275, −9.058722085716194789880205596641, −8.926648884687185212925284558356, −8.226901402151902866738141356653, −8.064326474338784466073449581357, −7.64062808339153370273001563858, −7.42891881658125989292916077155, −6.57419690303017447639645291802, −6.48770660992820850167625129137, −5.46364590272685055679858231450, −5.20380729829409182010252153007, −4.68923333742045961117131458509, −4.17257174932616932640189191169, −3.63602814170657235871932812636, −3.07811712083300008315310753009, −2.52872083637152739966316508117, −1.73120086306290826165577023814, 0, 0, 1.73120086306290826165577023814, 2.52872083637152739966316508117, 3.07811712083300008315310753009, 3.63602814170657235871932812636, 4.17257174932616932640189191169, 4.68923333742045961117131458509, 5.20380729829409182010252153007, 5.46364590272685055679858231450, 6.48770660992820850167625129137, 6.57419690303017447639645291802, 7.42891881658125989292916077155, 7.64062808339153370273001563858, 8.064326474338784466073449581357, 8.226901402151902866738141356653, 8.926648884687185212925284558356, 9.058722085716194789880205596641, 10.14648392519205838162628278275, 10.15868904826550110521612182023

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