Properties

Label 4-720e2-1.1-c1e2-0-16
Degree $4$
Conductor $518400$
Sign $1$
Analytic cond. $33.0536$
Root an. cond. $2.39775$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·19-s + 25-s − 12·29-s + 4·43-s − 12·47-s − 4·49-s + 12·53-s + 4·67-s + 12·71-s + 4·73-s + 16·97-s + 24·101-s + 8·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 4·169-s + 173-s + 179-s + 181-s + 191-s + ⋯
L(s)  = 1  + 0.917·19-s + 1/5·25-s − 2.22·29-s + 0.609·43-s − 1.75·47-s − 4/7·49-s + 1.64·53-s + 0.488·67-s + 1.42·71-s + 0.468·73-s + 1.62·97-s + 2.38·101-s + 8/11·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 0.307·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-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: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 518400,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.717666487\)
\(L(\frac12)\) \(\approx\) \(1.717666487\)
\(L(\frac{3}{2})\) 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_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good7$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - 8 T^{2} + p^{2} T^{4} \)
13$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
37$C_2^2$ \( 1 - 20 T^{2} + p^{2} T^{4} \)
41$C_2^2$ \( 1 - 8 T^{2} + p^{2} T^{4} \)
43$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
47$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
53$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + p T^{2} ) \)
59$C_2^2$ \( 1 + 64 T^{2} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 94 T^{2} + p^{2} T^{4} \)
67$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
71$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + p T^{2} ) \)
73$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
79$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \)
83$C_2^2$ \( 1 + 34 T^{2} + p^{2} T^{4} \)
89$C_2^2$ \( 1 + 136 T^{2} + p^{2} T^{4} \)
97$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 - 2 T + p T^{2} ) \)
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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

−8.570505404506489365485194754681, −7.86395387935307144457668537480, −7.68212415011676003241923904488, −7.19326217043332633518885405286, −6.69142412572434239264966491246, −6.21929547162502900344058073574, −5.62349690304331216178446997419, −5.28303168521316087694117889803, −4.79522516051746724345708917872, −4.11460828351980967913724777536, −3.53802008148141228672882166493, −3.19612235184291826155664884330, −2.27340870190227540273109142871, −1.75259533280598802269542867052, −0.69305652271597281137165976363, 0.69305652271597281137165976363, 1.75259533280598802269542867052, 2.27340870190227540273109142871, 3.19612235184291826155664884330, 3.53802008148141228672882166493, 4.11460828351980967913724777536, 4.79522516051746724345708917872, 5.28303168521316087694117889803, 5.62349690304331216178446997419, 6.21929547162502900344058073574, 6.69142412572434239264966491246, 7.19326217043332633518885405286, 7.68212415011676003241923904488, 7.86395387935307144457668537480, 8.570505404506489365485194754681

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