Properties

Label 4-768e2-1.1-c3e2-0-14
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 $0$

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: \(0\)
Selberg data: \((4,\ 589824,\ (\ :3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(1.514896681\)
\(L(\frac12)\) \(\approx\) \(1.514896681\)
\(L(\frac{5}{2})\) 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

−10.14954254829679674780230747533, −9.498111105333350892084779592978, −9.214618776931419988546075362463, −9.052093973087600044773336348302, −8.398935249589196886577598331979, −8.398805403693490735520427270585, −7.18970957262973087806883537088, −6.95812402953738979044416973864, −6.66585938539681730376616789586, −6.47715174776521429248282211237, −5.84910501489744073833959044720, −5.08333347813586004880108509639, −4.61504904368624792666605264565, −4.46602022798765925081095274529, −3.35154868152655505735652216512, −3.21528388210715666629842769754, −2.56142429047478835611354208653, −2.13195738594833807605148774066, −0.905440116914145266160422663682, −0.41852356046069948294231679342, 0.41852356046069948294231679342, 0.905440116914145266160422663682, 2.13195738594833807605148774066, 2.56142429047478835611354208653, 3.21528388210715666629842769754, 3.35154868152655505735652216512, 4.46602022798765925081095274529, 4.61504904368624792666605264565, 5.08333347813586004880108509639, 5.84910501489744073833959044720, 6.47715174776521429248282211237, 6.66585938539681730376616789586, 6.95812402953738979044416973864, 7.18970957262973087806883537088, 8.398805403693490735520427270585, 8.398935249589196886577598331979, 9.052093973087600044773336348302, 9.214618776931419988546075362463, 9.498111105333350892084779592978, 10.14954254829679674780230747533

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