Properties

Label 4-768e2-1.1-c3e2-0-34
Degree $4$
Conductor $589824$
Sign $1$
Analytic cond. $2053.31$
Root an. cond. $6.73152$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 32·7-s − 9·9-s − 252·17-s − 336·23-s + 214·25-s − 176·31-s − 84·41-s − 192·47-s + 82·49-s − 288·63-s − 1.58e3·71-s − 436·73-s − 1.04e3·79-s + 81·81-s − 1.62e3·89-s + 2.30e3·97-s − 256·103-s − 924·113-s − 8.06e3·119-s + 2.51e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 2.26e3·153-s + ⋯
L(s)  = 1  + 1.72·7-s − 1/3·9-s − 3.59·17-s − 3.04·23-s + 1.71·25-s − 1.01·31-s − 0.319·41-s − 0.595·47-s + 0.239·49-s − 0.575·63-s − 2.64·71-s − 0.699·73-s − 1.48·79-s + 1/9·81-s − 1.92·89-s + 2.41·97-s − 0.244·103-s − 0.769·113-s − 6.21·119-s + 1.89·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 1.19·153-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 589824 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 589824 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(589824\)    =    \(2^{16} \cdot 3^{2}\)
Sign: $1$
Analytic conductor: \(2053.31\)
Root analytic conductor: \(6.73152\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 589824,\ (\ :3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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$C_2$ \( 1 + p^{2} T^{2} \)
good5$C_2^2$ \( 1 - 214 T^{2} + p^{6} T^{4} \)
7$C_2$ \( ( 1 - 16 T + p^{3} T^{2} )^{2} \)
11$C_2^2$ \( 1 - 2518 T^{2} + p^{6} T^{4} \)
13$C_2^2$ \( 1 - 2950 T^{2} + p^{6} T^{4} \)
17$C_2$ \( ( 1 + 126 T + p^{3} T^{2} )^{2} \)
19$C_2^2$ \( 1 - 13318 T^{2} + p^{6} T^{4} \)
23$C_2$ \( ( 1 + 168 T + p^{3} T^{2} )^{2} \)
29$C_2^2$ \( 1 - 47878 T^{2} + p^{6} T^{4} \)
31$C_2$ \( ( 1 + 88 T + p^{3} T^{2} )^{2} \)
37$C_2^2$ \( 1 - 36790 T^{2} + p^{6} T^{4} \)
41$C_2$ \( ( 1 + 42 T + p^{3} T^{2} )^{2} \)
43$C_2^2$ \( 1 - 156310 T^{2} + p^{6} T^{4} \)
47$C_2$ \( ( 1 + 96 T + p^{3} T^{2} )^{2} \)
53$C_2^2$ \( 1 - 258550 T^{2} + p^{6} T^{4} \)
59$C_2^2$ \( 1 + 24842 T^{2} + p^{6} T^{4} \)
61$C_2^2$ \( 1 - 164518 T^{2} + p^{6} T^{4} \)
67$C_2^2$ \( 1 + 179930 T^{2} + p^{6} T^{4} \)
71$C_2$ \( ( 1 + 792 T + p^{3} T^{2} )^{2} \)
73$C_2$ \( ( 1 + 218 T + p^{3} T^{2} )^{2} \)
79$C_2$ \( ( 1 + 520 T + p^{3} T^{2} )^{2} \)
83$C_2^2$ \( 1 - 901510 T^{2} + p^{6} T^{4} \)
89$C_2$ \( ( 1 + 810 T + p^{3} T^{2} )^{2} \)
97$C_2$ \( ( 1 - 1154 T + p^{3} T^{2} )^{2} \)
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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

−9.728764591749159091974519878510, −9.045355283944169556423918986879, −8.701533324920314270409308117076, −8.635527349977539747011059829983, −8.022020101694580846596646665026, −7.71842667830765278883420758693, −6.98699398808047146551979605127, −6.79607051684719350040688854980, −6.04759582014127750420073460243, −5.89490957806419267866056687711, −4.92668916845336885518751298676, −4.80812291537779864572969360517, −4.22693213939216774204977394023, −4.00223952033311384578301692134, −2.96440634633205244086595789393, −2.32314795007014980898382008473, −1.86583473427032091233652659999, −1.47882309135144815357803456528, 0, 0, 1.47882309135144815357803456528, 1.86583473427032091233652659999, 2.32314795007014980898382008473, 2.96440634633205244086595789393, 4.00223952033311384578301692134, 4.22693213939216774204977394023, 4.80812291537779864572969360517, 4.92668916845336885518751298676, 5.89490957806419267866056687711, 6.04759582014127750420073460243, 6.79607051684719350040688854980, 6.98699398808047146551979605127, 7.71842667830765278883420758693, 8.022020101694580846596646665026, 8.635527349977539747011059829983, 8.701533324920314270409308117076, 9.045355283944169556423918986879, 9.728764591749159091974519878510

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