Properties

Label 4-840e2-1.1-c1e2-0-56
Degree $4$
Conductor $705600$
Sign $-1$
Analytic cond. $44.9896$
Root an. cond. $2.58987$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $1$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 9-s − 8·11-s + 25-s − 4·29-s − 4·37-s + 24·43-s − 7·49-s + 12·53-s + 8·67-s + 16·71-s − 16·79-s + 81-s − 8·99-s − 8·107-s − 36·109-s − 12·113-s + 26·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 10·169-s + ⋯
L(s)  = 1  + 1/3·9-s − 2.41·11-s + 1/5·25-s − 0.742·29-s − 0.657·37-s + 3.65·43-s − 49-s + 1.64·53-s + 0.977·67-s + 1.89·71-s − 1.80·79-s + 1/9·81-s − 0.804·99-s − 0.773·107-s − 3.44·109-s − 1.12·113-s + 2.36·121-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 + 0.769·169-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 705600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 705600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(705600\)    =    \(2^{6} \cdot 3^{2} \cdot 5^{2} \cdot 7^{2}\)
Sign: $-1$
Analytic conductor: \(44.9896\)
Root analytic conductor: \(2.58987\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 705600,\ (\ :1/2, 1/2),\ -1)\)

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$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
5$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
7$C_2$ \( 1 + p T^{2} \)
good11$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
37$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
43$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
61$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
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$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
97$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
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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

−8.099153868875425746869132395971, −7.66802799727824255195681387677, −7.28172218841942605198911351885, −6.89270810769313104844169224865, −6.25607757020835770318385102643, −5.59585158632682227968586342616, −5.39353249496950172428782136456, −5.03583353342906789625335566110, −4.21697652095783996379508576174, −3.95303834850677407956821293662, −3.10051698585138296427814999448, −2.50557638324841966716258912254, −2.25002432004047770108316173263, −1.09793012226940698545121907769, 0, 1.09793012226940698545121907769, 2.25002432004047770108316173263, 2.50557638324841966716258912254, 3.10051698585138296427814999448, 3.95303834850677407956821293662, 4.21697652095783996379508576174, 5.03583353342906789625335566110, 5.39353249496950172428782136456, 5.59585158632682227968586342616, 6.25607757020835770318385102643, 6.89270810769313104844169224865, 7.28172218841942605198911351885, 7.66802799727824255195681387677, 8.099153868875425746869132395971

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