Properties

Label 4-2475e2-1.1-c0e2-0-1
Degree $4$
Conductor $6125625$
Sign $1$
Analytic cond. $1.52568$
Root an. cond. $1.11138$
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  + 3-s − 4-s − 11-s − 12-s + 2·23-s − 27-s + 31-s − 33-s + 2·37-s + 44-s − 47-s − 49-s + 2·53-s + 59-s + 64-s − 67-s + 2·69-s − 2·71-s − 81-s + 4·89-s − 2·92-s + 93-s − 97-s − 103-s + 108-s + 2·111-s − 113-s + ⋯
L(s)  = 1  + 3-s − 4-s − 11-s − 12-s + 2·23-s − 27-s + 31-s − 33-s + 2·37-s + 44-s − 47-s − 49-s + 2·53-s + 59-s + 64-s − 67-s + 2·69-s − 2·71-s − 81-s + 4·89-s − 2·92-s + 93-s − 97-s − 103-s + 108-s + 2·111-s − 113-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6125625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6125625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(6125625\)    =    \(3^{4} \cdot 5^{4} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(1.52568\)
Root analytic conductor: \(1.11138\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 6125625,\ (\ :0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.201706605\)
\(L(\frac12)\) \(\approx\) \(1.201706605\)
\(L(1)\) 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)$
bad3$C_2$ \( 1 - T + T^{2} \)
5 \( 1 \)
11$C_2$ \( 1 + T + T^{2} \)
good2$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
7$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
13$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
17$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
19$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
23$C_2$ \( ( 1 - T + T^{2} )^{2} \)
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$C_2$ \( ( 1 - T + T^{2} )^{2} \)
41$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
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_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 - T + T^{2} ) \)
71$C_2$ \( ( 1 + T + T^{2} )^{2} \)
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_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 - T + T^{2} ) \)
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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

−9.410985929848712615106790997803, −8.758412589084946704168195073635, −8.701952470505225732289211597338, −8.223371281512659741714185689608, −7.88052780456012402018296595116, −7.59522228250637489045989516451, −7.07865715415286509771607565017, −6.77191794842081015367782634723, −6.01732056864407023036729448778, −5.94360258333511489418202447152, −5.13858760138922981260008018134, −4.92364133138918527906150437846, −4.68314943332594180782099277072, −3.96925233145204346311140761045, −3.73230311150702403781378349087, −3.00898745401341516618974520873, −2.70152505316010835054888682479, −2.43856473416329474941547616601, −1.52901858271369773888075768574, −0.72137328655715355456477280561, 0.72137328655715355456477280561, 1.52901858271369773888075768574, 2.43856473416329474941547616601, 2.70152505316010835054888682479, 3.00898745401341516618974520873, 3.73230311150702403781378349087, 3.96925233145204346311140761045, 4.68314943332594180782099277072, 4.92364133138918527906150437846, 5.13858760138922981260008018134, 5.94360258333511489418202447152, 6.01732056864407023036729448778, 6.77191794842081015367782634723, 7.07865715415286509771607565017, 7.59522228250637489045989516451, 7.88052780456012402018296595116, 8.223371281512659741714185689608, 8.701952470505225732289211597338, 8.758412589084946704168195073635, 9.410985929848712615106790997803

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