Properties

Label 4-2592e2-1.1-c1e2-0-1
Degree $4$
Conductor $6718464$
Sign $1$
Analytic cond. $428.375$
Root an. cond. $4.54942$
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  + 2·5-s − 7-s + 2·11-s − 13-s − 12·17-s − 10·19-s − 6·23-s + 5·25-s + 8·29-s − 8·31-s − 2·35-s − 10·37-s + 8·41-s + 4·43-s + 10·47-s + 7·49-s − 8·53-s + 4·55-s − 14·59-s − 3·61-s − 2·65-s + 13·67-s − 8·71-s + 18·73-s − 2·77-s − 11·79-s + 12·83-s + ⋯
L(s)  = 1  + 0.894·5-s − 0.377·7-s + 0.603·11-s − 0.277·13-s − 2.91·17-s − 2.29·19-s − 1.25·23-s + 25-s + 1.48·29-s − 1.43·31-s − 0.338·35-s − 1.64·37-s + 1.24·41-s + 0.609·43-s + 1.45·47-s + 49-s − 1.09·53-s + 0.539·55-s − 1.82·59-s − 0.384·61-s − 0.248·65-s + 1.58·67-s − 0.949·71-s + 2.10·73-s − 0.227·77-s − 1.23·79-s + 1.31·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6718464 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6718464 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(6718464\)    =    \(2^{10} \cdot 3^{8}\)
Sign: $1$
Analytic conductor: \(428.375\)
Root analytic conductor: \(4.54942\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{2592} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 6718464,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8392769240\)
\(L(\frac12)\) \(\approx\) \(0.8392769240\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good5$C_2^2$ \( 1 - 2 T - T^{2} - 2 p T^{3} + p^{2} T^{4} \)
7$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
11$C_2^2$ \( 1 - 2 T - 7 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
13$C_2^2$ \( 1 + T - 12 T^{2} + p T^{3} + p^{2} T^{4} \)
17$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 + 6 T + 13 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 - 8 T + 35 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
31$C_2^2$ \( 1 + 8 T + 33 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
37$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 - 8 T + 23 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
43$C_2^2$ \( 1 - 4 T - 27 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 10 T + 53 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
53$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 + 14 T + 137 T^{2} + 14 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 3 T - 52 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 - 13 T + 102 T^{2} - 13 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \)
79$C_2^2$ \( 1 + 11 T + 42 T^{2} + 11 p T^{3} + p^{2} T^{4} \)
83$C_2^2$ \( 1 - 12 T + 61 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 + T - 96 T^{2} + p T^{3} + p^{2} T^{4} \)
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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

−9.009022716472264418779033159351, −8.649944092255145522756546529039, −8.634537837675451359002650181598, −8.046976466648692633894050132124, −7.44434627837536520937000682171, −6.90399314407430787810710999240, −6.82889127460352438549083489107, −6.33343729755726719893649220422, −6.04797544319859575536319676420, −5.87392053169782770705470233369, −5.05229215526127219463454538780, −4.59009484245182620070375389666, −4.45531722659863800064174109647, −3.89219501947325747785986371395, −3.54820953935455949684083733370, −2.58888551232916792624614062988, −2.20401594086502327641214219011, −2.19215591558151794541522193288, −1.38136835743757724794755140731, −0.28553088501887923982534574652, 0.28553088501887923982534574652, 1.38136835743757724794755140731, 2.19215591558151794541522193288, 2.20401594086502327641214219011, 2.58888551232916792624614062988, 3.54820953935455949684083733370, 3.89219501947325747785986371395, 4.45531722659863800064174109647, 4.59009484245182620070375389666, 5.05229215526127219463454538780, 5.87392053169782770705470233369, 6.04797544319859575536319676420, 6.33343729755726719893649220422, 6.82889127460352438549083489107, 6.90399314407430787810710999240, 7.44434627837536520937000682171, 8.046976466648692633894050132124, 8.634537837675451359002650181598, 8.649944092255145522756546529039, 9.009022716472264418779033159351

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