Properties

Label 4-240e2-1.1-c5e2-0-0
Degree $4$
Conductor $57600$
Sign $1$
Analytic cond. $1481.64$
Root an. cond. $6.20420$
Motivic weight $5$
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  − 110·5-s − 81·9-s + 1.00e3·11-s − 2.68e3·19-s + 8.97e3·25-s + 5.29e3·29-s + 1.12e4·31-s − 3.79e4·41-s + 8.91e3·45-s + 3.35e4·49-s − 1.10e5·55-s − 5.66e4·59-s + 3.65e4·61-s + 5.76e4·71-s + 1.20e5·79-s + 6.56e3·81-s − 4.53e4·89-s + 2.95e5·95-s − 8.10e4·99-s + 3.35e5·101-s + 1.07e5·109-s + 4.27e5·121-s − 6.43e5·125-s + 127-s + 131-s + 137-s + 139-s + ⋯
L(s)  = 1  − 1.96·5-s − 1/3·9-s + 2.49·11-s − 1.70·19-s + 2.87·25-s + 1.16·29-s + 2.09·31-s − 3.52·41-s + 0.655·45-s + 1.99·49-s − 4.90·55-s − 2.11·59-s + 1.25·61-s + 1.35·71-s + 2.17·79-s + 1/9·81-s − 0.606·89-s + 3.36·95-s − 0.830·99-s + 3.27·101-s + 0.867·109-s + 2.65·121-s − 3.68·125-s + 5.50e−6·127-s + 5.09e−6·131-s + 4.55e−6·137-s + 4.38e−6·139-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 57600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 57600 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(57600\)    =    \(2^{8} \cdot 3^{2} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(1481.64\)
Root analytic conductor: \(6.20420\)
Motivic weight: \(5\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 57600,\ (\ :5/2, 5/2),\ 1)\)

Particular Values

\(L(3)\) \(\approx\) \(1.908059450\)
\(L(\frac12)\) \(\approx\) \(1.908059450\)
\(L(\frac{7}{2})\) 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$C_2$ \( 1 + p^{4} T^{2} \)
5$C_2$ \( 1 + 22 p T + p^{5} T^{2} \)
good7$C_2^2$ \( 1 - 33598 T^{2} + p^{10} T^{4} \)
11$C_2$ \( ( 1 - 500 T + p^{5} T^{2} )^{2} \)
13$C_2^2$ \( 1 - 659642 T^{2} + p^{10} T^{4} \)
17$C_2^2$ \( 1 - 541458 T^{2} + p^{10} T^{4} \)
19$C_2$ \( ( 1 + 1344 T + p^{5} T^{2} )^{2} \)
23$C_2^2$ \( 1 + 3937314 T^{2} + p^{10} T^{4} \)
29$C_2$ \( ( 1 - 2646 T + p^{5} T^{2} )^{2} \)
31$C_2$ \( ( 1 - 5612 T + p^{5} T^{2} )^{2} \)
37$C_2^2$ \( 1 - 85572970 T^{2} + p^{10} T^{4} \)
41$C_2$ \( ( 1 + 18986 T + p^{5} T^{2} )^{2} \)
43$C_2^2$ \( 1 - 288237670 T^{2} + p^{10} T^{4} \)
47$C_2^2$ \( 1 - 379480014 T^{2} + p^{10} T^{4} \)
53$C_2^2$ \( 1 + 747967430 T^{2} + p^{10} T^{4} \)
59$C_2$ \( ( 1 + 28300 T + p^{5} T^{2} )^{2} \)
61$C_2$ \( ( 1 - 18290 T + p^{5} T^{2} )^{2} \)
67$C_2^2$ \( 1 + 1649943722 T^{2} + p^{10} T^{4} \)
71$C_2$ \( ( 1 - 28800 T + p^{5} T^{2} )^{2} \)
73$C_2^2$ \( 1 - 3197010322 T^{2} + p^{10} T^{4} \)
79$C_2$ \( ( 1 - 60228 T + p^{5} T^{2} )^{2} \)
83$C_2^2$ \( 1 - 7871990262 T^{2} + p^{10} T^{4} \)
89$C_2$ \( ( 1 + 22678 T + p^{5} T^{2} )^{2} \)
97$C_2^2$ \( 1 - 15808047490 T^{2} + p^{10} T^{4} \)
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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

−11.69782167442206144775883138861, −11.20747081541582643734793956714, −10.55941209883342872718618195504, −10.26010291015012095007904221638, −9.473144008617710720752269041042, −8.710312219705324126293446320646, −8.643658348492452052289983702555, −8.262108870725686046623952956376, −7.58730004858450567929147851818, −6.84204660766136714508617821745, −6.48634777799783507503717016141, −6.37020272512917050035432909804, −5.03796778033828196776290972658, −4.54012922107060818629922820693, −4.08579894411393318795105142425, −3.59843393810766853176200248235, −3.05244430074198062550540727309, −1.98549572103489739987571344828, −1.00743887055576110195755109867, −0.50279250293168393462702627084, 0.50279250293168393462702627084, 1.00743887055576110195755109867, 1.98549572103489739987571344828, 3.05244430074198062550540727309, 3.59843393810766853176200248235, 4.08579894411393318795105142425, 4.54012922107060818629922820693, 5.03796778033828196776290972658, 6.37020272512917050035432909804, 6.48634777799783507503717016141, 6.84204660766136714508617821745, 7.58730004858450567929147851818, 8.262108870725686046623952956376, 8.643658348492452052289983702555, 8.710312219705324126293446320646, 9.473144008617710720752269041042, 10.26010291015012095007904221638, 10.55941209883342872718618195504, 11.20747081541582643734793956714, 11.69782167442206144775883138861

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