Properties

Label 4-768e2-1.1-c1e2-0-29
Degree $4$
Conductor $589824$
Sign $1$
Analytic cond. $37.6076$
Root an. cond. $2.47639$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·5-s + 9-s + 8·13-s − 4·17-s + 2·25-s + 12·29-s + 16·37-s − 4·41-s + 4·45-s − 10·49-s + 12·53-s + 32·65-s + 12·73-s + 81-s − 16·85-s + 20·89-s − 4·97-s − 20·101-s − 8·109-s − 12·113-s + 8·117-s − 22·121-s − 28·125-s + 127-s + 131-s + 137-s + 139-s + ⋯
L(s)  = 1  + 1.78·5-s + 1/3·9-s + 2.21·13-s − 0.970·17-s + 2/5·25-s + 2.22·29-s + 2.63·37-s − 0.624·41-s + 0.596·45-s − 1.42·49-s + 1.64·53-s + 3.96·65-s + 1.40·73-s + 1/9·81-s − 1.73·85-s + 2.11·89-s − 0.406·97-s − 1.99·101-s − 0.766·109-s − 1.12·113-s + 0.739·117-s − 2·121-s − 2.50·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯

Functional equation

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

Invariants

Degree: \(4\)
Conductor: \(589824\)    =    \(2^{16} \cdot 3^{2}\)
Sign: $1$
Analytic conductor: \(37.6076\)
Root analytic conductor: \(2.47639\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 589824,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.439859856\)
\(L(\frac12)\) \(\approx\) \(3.439859856\)
\(L(\frac{3}{2})\) 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$ \( ( 1 - T )( 1 + T ) \)
good5$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
7$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
11$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
23$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
47$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
61$C_2$ \( ( 1 + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
71$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
73$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
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

−8.420702100846206882694460804856, −8.137356992201587112320496664394, −7.60982656432259660497620212594, −6.68785915312412950432661260762, −6.53729896264740462444493449031, −6.03168549170604009889878923550, −6.02952025279743756519610804548, −5.12505772656764585053702097555, −4.84641162437662246944769643988, −3.87678607648250663694375372790, −3.87120361857012739887619384713, −2.63713724931622987423343237649, −2.47902910741521785662962015803, −1.51663712403377833666587682944, −1.09527349773841298009626376824, 1.09527349773841298009626376824, 1.51663712403377833666587682944, 2.47902910741521785662962015803, 2.63713724931622987423343237649, 3.87120361857012739887619384713, 3.87678607648250663694375372790, 4.84641162437662246944769643988, 5.12505772656764585053702097555, 6.02952025279743756519610804548, 6.03168549170604009889878923550, 6.53729896264740462444493449031, 6.68785915312412950432661260762, 7.60982656432259660497620212594, 8.137356992201587112320496664394, 8.420702100846206882694460804856

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