Properties

Label 4-30e4-1.1-c2e2-0-10
Degree $4$
Conductor $810000$
Sign $1$
Analytic cond. $601.388$
Root an. cond. $4.95209$
Motivic weight $2$
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-s − 3·4-s − 7·8-s + 5·16-s + 28·17-s + 33·32-s + 28·34-s + 98·49-s + 172·53-s + 236·61-s + 13·64-s − 84·68-s + 98·98-s + 172·106-s + 44·109-s − 412·113-s + 242·121-s + 236·122-s + 127-s − 119·128-s + 131-s − 196·136-s + 137-s + 139-s + 149-s + 151-s + 157-s + ⋯
L(s)  = 1  + 1/2·2-s − 3/4·4-s − 7/8·8-s + 5/16·16-s + 1.64·17-s + 1.03·32-s + 0.823·34-s + 2·49-s + 3.24·53-s + 3.86·61-s + 0.203·64-s − 1.23·68-s + 98-s + 1.62·106-s + 0.403·109-s − 3.64·113-s + 2·121-s + 1.93·122-s + 0.00787·127-s − 0.929·128-s + 0.00763·131-s − 1.44·136-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 810000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 810000 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(810000\)    =    \(2^{4} \cdot 3^{4} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(601.388\)
Root analytic conductor: \(4.95209\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{900} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 810000,\ (\ :1, 1),\ 1)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(3.056025899\)
\(L(\frac12)\) \(\approx\) \(3.056025899\)
\(L(2)\) 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$C_2$ \( 1 - T + p^{2} T^{2} \)
3 \( 1 \)
5 \( 1 \)
good7$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
11$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
13$C_2$ \( ( 1 + p^{2} T^{2} )^{2} \)
17$C_2$ \( ( 1 - 14 T + p^{2} T^{2} )^{2} \)
19$C_2$ \( ( 1 - 22 T + p^{2} T^{2} )( 1 + 22 T + p^{2} T^{2} ) \)
23$C_2$ \( ( 1 - 34 T + p^{2} T^{2} )( 1 + 34 T + p^{2} T^{2} ) \)
29$C_2$ \( ( 1 + p^{2} T^{2} )^{2} \)
31$C_2$ \( ( 1 - 2 T + p^{2} T^{2} )( 1 + 2 T + p^{2} T^{2} ) \)
37$C_2$ \( ( 1 + p^{2} T^{2} )^{2} \)
41$C_2$ \( ( 1 + p^{2} T^{2} )^{2} \)
43$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
47$C_2$ \( ( 1 - 14 T + p^{2} T^{2} )( 1 + 14 T + p^{2} T^{2} ) \)
53$C_2$ \( ( 1 - 86 T + p^{2} T^{2} )^{2} \)
59$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
61$C_2$ \( ( 1 - 118 T + p^{2} T^{2} )^{2} \)
67$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
71$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
73$C_2$ \( ( 1 + p^{2} T^{2} )^{2} \)
79$C_2$ \( ( 1 - 98 T + p^{2} T^{2} )( 1 + 98 T + p^{2} T^{2} ) \)
83$C_2$ \( ( 1 - 154 T + p^{2} T^{2} )( 1 + 154 T + p^{2} T^{2} ) \)
89$C_2$ \( ( 1 + p^{2} T^{2} )^{2} \)
97$C_2$ \( ( 1 + p^{2} T^{2} )^{2} \)
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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

−9.891384690975723758663312508526, −9.864473179883106180022424344895, −9.364562297957143705828456821191, −8.769321032484791338039519622630, −8.426564959904299688452739566144, −8.227685420935828999900138046427, −7.52689278994781506594991910252, −7.10490609286310707001461012683, −6.80250495111581684565540076251, −5.97865886918113664478881843963, −5.55168824246522250293868340000, −5.48103353130138898002890199466, −4.88701795314231003894735663404, −4.17319792296347694424780866894, −3.82749204806894224196570705590, −3.47715187727858018604461441177, −2.68617202105089289588207724228, −2.22972943008531047637106077038, −1.05562944383288945052537382242, −0.65790968067746589783569440519, 0.65790968067746589783569440519, 1.05562944383288945052537382242, 2.22972943008531047637106077038, 2.68617202105089289588207724228, 3.47715187727858018604461441177, 3.82749204806894224196570705590, 4.17319792296347694424780866894, 4.88701795314231003894735663404, 5.48103353130138898002890199466, 5.55168824246522250293868340000, 5.97865886918113664478881843963, 6.80250495111581684565540076251, 7.10490609286310707001461012683, 7.52689278994781506594991910252, 8.227685420935828999900138046427, 8.426564959904299688452739566144, 8.769321032484791338039519622630, 9.364562297957143705828456821191, 9.864473179883106180022424344895, 9.891384690975723758663312508526

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