Properties

Label 4-675e2-1.1-c1e2-0-0
Degree $4$
Conductor $455625$
Sign $1$
Analytic cond. $29.0510$
Root an. cond. $2.32161$
Motivic weight $1$
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 + 2·4-s − 3·7-s − 5·8-s − 2·11-s − 2·13-s + 3·14-s + 5·16-s + 8·17-s − 16·19-s + 2·22-s − 3·23-s + 2·26-s − 6·28-s − 29-s − 10·32-s − 8·34-s + 8·37-s + 16·38-s + 5·41-s − 8·43-s − 4·44-s + 3·46-s − 7·47-s + 7·49-s − 4·52-s − 4·53-s + ⋯
L(s)  = 1  − 0.707·2-s + 4-s − 1.13·7-s − 1.76·8-s − 0.603·11-s − 0.554·13-s + 0.801·14-s + 5/4·16-s + 1.94·17-s − 3.67·19-s + 0.426·22-s − 0.625·23-s + 0.392·26-s − 1.13·28-s − 0.185·29-s − 1.76·32-s − 1.37·34-s + 1.31·37-s + 2.59·38-s + 0.780·41-s − 1.21·43-s − 0.603·44-s + 0.442·46-s − 1.02·47-s + 49-s − 0.554·52-s − 0.549·53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 455625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 455625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(455625\)    =    \(3^{6} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(29.0510\)
Root analytic conductor: \(2.32161\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 455625,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3820116304\)
\(L(\frac12)\) \(\approx\) \(0.3820116304\)
\(L(\frac{3}{2})\) 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)$
bad3 \( 1 \)
5 \( 1 \)
good2$C_2^2$ \( 1 + T - T^{2} + p T^{3} + p^{2} T^{4} \)
7$C_2^2$ \( 1 + 3 T + 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 + 2 T - 7 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 + 3 T - 14 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + T - 28 T^{2} + p T^{3} + p^{2} T^{4} \)
31$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
37$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 - 5 T - 16 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
43$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 13 T + p T^{2} ) \)
47$C_2^2$ \( 1 + 7 T + 2 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
53$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 + 14 T + 137 T^{2} + 14 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 7 T - 12 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + 3 T - 58 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
79$C_2^2$ \( 1 - 6 T - 43 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
83$C_2^2$ \( 1 + 9 T - 2 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 15 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 - 2 T - 93 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
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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

−10.91405946265766374870851749679, −10.25808217177034498632273759870, −9.895183945586868705403190120068, −9.401494894108609174218915590874, −9.133939079565944715854203308098, −8.470043208626085190084086385835, −8.226927594636734235714688426153, −7.66024324288023754575593224679, −7.33722671160166898074880901519, −6.44951178544787877921830913337, −6.44108223885072474479195228379, −5.99612353950498737137371204888, −5.62169403098116426301867502272, −4.68635528239629710217673204112, −4.20189868866273516923037271542, −3.22906221062972436043939803167, −3.15278084172945382871504386796, −2.32785361445130168953501196763, −1.85580140037294899991126995904, −0.34066087748587836479031662230, 0.34066087748587836479031662230, 1.85580140037294899991126995904, 2.32785361445130168953501196763, 3.15278084172945382871504386796, 3.22906221062972436043939803167, 4.20189868866273516923037271542, 4.68635528239629710217673204112, 5.62169403098116426301867502272, 5.99612353950498737137371204888, 6.44108223885072474479195228379, 6.44951178544787877921830913337, 7.33722671160166898074880901519, 7.66024324288023754575593224679, 8.226927594636734235714688426153, 8.470043208626085190084086385835, 9.133939079565944715854203308098, 9.401494894108609174218915590874, 9.895183945586868705403190120068, 10.25808217177034498632273759870, 10.91405946265766374870851749679

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