Properties

Label 4-3240e2-1.1-c0e2-0-17
Degree $4$
Conductor $10497600$
Sign $1$
Analytic cond. $2.61459$
Root an. cond. $1.27160$
Motivic weight $0$
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  + 2·2-s + 3·4-s + 5-s + 4·8-s + 2·10-s + 5·16-s + 2·19-s + 3·20-s − 23-s − 3·31-s + 6·32-s + 4·38-s + 4·40-s − 2·46-s − 2·47-s + 49-s − 2·53-s + 3·61-s − 6·62-s + 7·64-s + 6·76-s − 3·79-s + 5·80-s − 3·83-s − 3·92-s − 4·94-s + 2·95-s + ⋯
L(s)  = 1  + 2·2-s + 3·4-s + 5-s + 4·8-s + 2·10-s + 5·16-s + 2·19-s + 3·20-s − 23-s − 3·31-s + 6·32-s + 4·38-s + 4·40-s − 2·46-s − 2·47-s + 49-s − 2·53-s + 3·61-s − 6·62-s + 7·64-s + 6·76-s − 3·79-s + 5·80-s − 3·83-s − 3·92-s − 4·94-s + 2·95-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 10497600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10497600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(10497600\)    =    \(2^{6} \cdot 3^{8} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(2.61459\)
Root analytic conductor: \(1.27160\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 10497600,\ (\ :0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(7.279079742\)
\(L(\frac12)\) \(\approx\) \(7.279079742\)
\(L(1)\) 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$C_1$ \( ( 1 - T )^{2} \)
3 \( 1 \)
5$C_2$ \( 1 - T + T^{2} \)
good7$C_2^2$ \( 1 - T^{2} + T^{4} \)
11$C_2^2$ \( 1 - T^{2} + T^{4} \)
13$C_2^2$ \( 1 - T^{2} + T^{4} \)
17$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
19$C_2$ \( ( 1 - T + T^{2} )^{2} \)
23$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 - T + T^{2} ) \)
29$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
31$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T + T^{2} ) \)
37$C_2$ \( ( 1 + T^{2} )^{2} \)
41$C_2^2$ \( 1 - T^{2} + T^{4} \)
43$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
47$C_2$ \( ( 1 + T + T^{2} )^{2} \)
53$C_2$ \( ( 1 + T + T^{2} )^{2} \)
59$C_2^2$ \( 1 - T^{2} + T^{4} \)
61$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 - T + T^{2} ) \)
67$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
71$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
73$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
79$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T + T^{2} ) \)
83$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T + T^{2} ) \)
89$C_2$ \( ( 1 + T^{2} )^{2} \)
97$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
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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

−8.985415510924040841652380932550, −8.644431530835748219528419054636, −8.013046953223404996845233618761, −7.71548024687354504338870134728, −7.22920137302163859514935722425, −7.19342726863504151283290926730, −6.58598200328578019443890668890, −6.27324337157929223761397343478, −5.75645420659034615353133038412, −5.57426773100597854767857035521, −5.15994669509305035617800269702, −5.13865282203846834763251167716, −4.25294712941792186571947289940, −4.02295480227100309543703328782, −3.55573528832098950261198487490, −3.12388743892840018489691919905, −2.74207081252684412975609324357, −2.15725181972387401659466000115, −1.56808751671950202533415454776, −1.46879081297093408809818655709, 1.46879081297093408809818655709, 1.56808751671950202533415454776, 2.15725181972387401659466000115, 2.74207081252684412975609324357, 3.12388743892840018489691919905, 3.55573528832098950261198487490, 4.02295480227100309543703328782, 4.25294712941792186571947289940, 5.13865282203846834763251167716, 5.15994669509305035617800269702, 5.57426773100597854767857035521, 5.75645420659034615353133038412, 6.27324337157929223761397343478, 6.58598200328578019443890668890, 7.19342726863504151283290926730, 7.22920137302163859514935722425, 7.71548024687354504338870134728, 8.013046953223404996845233618761, 8.644431530835748219528419054636, 8.985415510924040841652380932550

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