Properties

Label 4-3240e2-1.1-c0e2-0-0
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

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 5-s − 7-s + 8-s + 10-s + 2·11-s − 13-s + 14-s − 16-s − 2·19-s − 2·22-s + 23-s + 26-s + 35-s − 4·37-s + 2·38-s − 40-s − 41-s − 46-s + 47-s + 49-s − 2·53-s − 2·55-s − 56-s − 59-s + 64-s + 65-s + ⋯
L(s)  = 1  − 2-s − 5-s − 7-s + 8-s + 10-s + 2·11-s − 13-s + 14-s − 16-s − 2·19-s − 2·22-s + 23-s + 26-s + 35-s − 4·37-s + 2·38-s − 40-s − 41-s − 46-s + 47-s + 49-s − 2·53-s − 2·55-s − 56-s − 59-s + 64-s + 65-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

\(L(\frac{1}{2})\) \(\approx\) \(0.1677165560\)
\(L(\frac12)\) \(\approx\) \(0.1677165560\)
\(L(1)\) 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_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 - T + T^{2} ) \)
11$C_2$ \( ( 1 - T + T^{2} )^{2} \)
13$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 - T + T^{2} ) \)
17$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
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$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
37$C_1$ \( ( 1 + T )^{4} \)
41$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 - T + T^{2} ) \)
43$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
47$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T + T^{2} ) \)
53$C_2$ \( ( 1 + T + T^{2} )^{2} \)
59$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 - T + T^{2} ) \)
61$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
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$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
83$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
89$C_1$ \( ( 1 + T )^{4} \)
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

−9.250648848181753343840710991336, −8.677963390654611546023830711172, −8.422186182482200567766248301543, −8.122059813772090837011813955465, −7.40954608538076134563481673061, −7.15489562445487458480781604003, −6.94895603657755259193624640769, −6.54809510236037711097521435193, −6.35755515385569860338094178747, −5.59159254068738686641822853341, −5.19779706484269263561797165010, −4.64139054334581828293505593926, −4.27755478882460657635391282139, −3.99523513629948252906578176199, −3.50679612418110058989646829639, −3.18867958611158477146406243618, −2.47154083436437042987522116496, −1.59632432546985261808095667094, −1.58439722650982509814668325634, −0.29806711929197286982440614761, 0.29806711929197286982440614761, 1.58439722650982509814668325634, 1.59632432546985261808095667094, 2.47154083436437042987522116496, 3.18867958611158477146406243618, 3.50679612418110058989646829639, 3.99523513629948252906578176199, 4.27755478882460657635391282139, 4.64139054334581828293505593926, 5.19779706484269263561797165010, 5.59159254068738686641822853341, 6.35755515385569860338094178747, 6.54809510236037711097521435193, 6.94895603657755259193624640769, 7.15489562445487458480781604003, 7.40954608538076134563481673061, 8.122059813772090837011813955465, 8.422186182482200567766248301543, 8.677963390654611546023830711172, 9.250648848181753343840710991336

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