Properties

Label 4-3240e2-1.1-c0e2-0-9
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-s + 5-s + 8-s − 10-s − 16-s + 2·19-s − 23-s + 3·31-s − 2·38-s + 40-s + 46-s − 2·47-s + 49-s − 2·53-s − 3·61-s − 3·62-s + 64-s + 3·79-s − 80-s + 3·83-s + 2·94-s + 2·95-s − 98-s + 2·106-s − 115-s + 121-s + 3·122-s + ⋯
L(s)  = 1  − 2-s + 5-s + 8-s − 10-s − 16-s + 2·19-s − 23-s + 3·31-s − 2·38-s + 40-s + 46-s − 2·47-s + 49-s − 2·53-s − 3·61-s − 3·62-s + 64-s + 3·79-s − 80-s + 3·83-s + 2·94-s + 2·95-s − 98-s + 2·106-s − 115-s + 121-s + 3·122-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

\(L(\frac{1}{2})\) \(\approx\) \(0.9847161632\)
\(L(\frac12)\) \(\approx\) \(0.9847161632\)
\(L(1)\) 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$C_2$ \( 1 + T + T^{2} \)
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

−9.114380424932356413029279956265, −8.774639717621336695108187650658, −8.063613244514016110045635227022, −7.964293178622560157125849434103, −7.83330835762579070760500838464, −7.35701975827551266653767395344, −6.62366266728923752217915642105, −6.52888064506062083826810214654, −6.09157546690649956605185555807, −5.74538172411086929913050377280, −5.07868091854618054292975873012, −4.84980638737296705692982102553, −4.60308608784609179346475750161, −3.95428718467989718258389281024, −3.24176508697820678014290675531, −3.12133694147074461780703827854, −2.33271783431510951357366550408, −1.87788608412848438253145007340, −1.35224497254611265376290807971, −0.78878678362419557792982122383, 0.78878678362419557792982122383, 1.35224497254611265376290807971, 1.87788608412848438253145007340, 2.33271783431510951357366550408, 3.12133694147074461780703827854, 3.24176508697820678014290675531, 3.95428718467989718258389281024, 4.60308608784609179346475750161, 4.84980638737296705692982102553, 5.07868091854618054292975873012, 5.74538172411086929913050377280, 6.09157546690649956605185555807, 6.52888064506062083826810214654, 6.62366266728923752217915642105, 7.35701975827551266653767395344, 7.83330835762579070760500838464, 7.964293178622560157125849434103, 8.063613244514016110045635227022, 8.774639717621336695108187650658, 9.114380424932356413029279956265

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