Properties

Label 4-9280e2-1.1-c1e2-0-2
Degree $4$
Conductor $86118400$
Sign $1$
Analytic cond. $5490.98$
Root an. cond. $8.60820$
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  − 2·5-s − 6·9-s + 4·13-s − 12·17-s + 3·25-s − 2·29-s − 12·37-s + 20·41-s + 12·45-s − 6·49-s − 12·53-s − 28·61-s − 8·65-s + 20·73-s + 27·81-s + 24·85-s + 20·89-s + 4·97-s + 20·101-s − 28·109-s + 4·113-s − 24·117-s − 14·121-s − 4·125-s + 127-s + 131-s + 137-s + ⋯
L(s)  = 1  − 0.894·5-s − 2·9-s + 1.10·13-s − 2.91·17-s + 3/5·25-s − 0.371·29-s − 1.97·37-s + 3.12·41-s + 1.78·45-s − 6/7·49-s − 1.64·53-s − 3.58·61-s − 0.992·65-s + 2.34·73-s + 3·81-s + 2.60·85-s + 2.11·89-s + 0.406·97-s + 1.99·101-s − 2.68·109-s + 0.376·113-s − 2.21·117-s − 1.27·121-s − 0.357·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 86118400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 86118400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(86118400\)    =    \(2^{12} \cdot 5^{2} \cdot 29^{2}\)
Sign: $1$
Analytic conductor: \(5490.98\)
Root analytic conductor: \(8.60820\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 86118400,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.054309791\)
\(L(\frac12)\) \(\approx\) \(1.054309791\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5$C_1$ \( ( 1 + T )^{2} \)
29$C_1$ \( ( 1 + T )^{2} \)
good3$C_2$ \( ( 1 + p T^{2} )^{2} \)
7$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
19$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - 26 T^{2} + p^{2} T^{4} \)
31$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
37$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + p T^{2} )^{2} \)
47$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 + 86 T^{2} + p^{2} T^{4} \)
61$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 66 T^{2} + p^{2} T^{4} \)
71$C_2^2$ \( 1 + 110 T^{2} + p^{2} T^{4} \)
73$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
79$C_2^2$ \( 1 - 42 T^{2} + p^{2} T^{4} \)
83$C_2^2$ \( 1 + 94 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
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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

−7.915741192142770306796392337153, −7.65466416972411228108915109496, −7.25674017778992761562791212322, −6.76166504826643509111164057678, −6.32699741209488423379027461421, −6.28531981175074694388211665879, −6.05377776517651969832706831167, −5.46619921032959415498599999691, −5.04850756767225299707806332295, −4.80321004484362830805964343657, −4.30699306668048411916660210810, −4.11831036092683905676362463611, −3.48824334895869503502831328921, −3.37602417578065219049873219738, −2.71983392363809107114996002847, −2.65113376973902462734323442287, −1.80735310867175143682669337234, −1.77832685771979051739901999229, −0.59234524879977102547528012161, −0.37547912202389653735738308782, 0.37547912202389653735738308782, 0.59234524879977102547528012161, 1.77832685771979051739901999229, 1.80735310867175143682669337234, 2.65113376973902462734323442287, 2.71983392363809107114996002847, 3.37602417578065219049873219738, 3.48824334895869503502831328921, 4.11831036092683905676362463611, 4.30699306668048411916660210810, 4.80321004484362830805964343657, 5.04850756767225299707806332295, 5.46619921032959415498599999691, 6.05377776517651969832706831167, 6.28531981175074694388211665879, 6.32699741209488423379027461421, 6.76166504826643509111164057678, 7.25674017778992761562791212322, 7.65466416972411228108915109496, 7.915741192142770306796392337153

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