Properties

Label 4-675e2-1.1-c0e2-0-1
Degree $4$
Conductor $455625$
Sign $1$
Analytic cond. $0.113480$
Root an. cond. $0.580404$
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  + 4-s + 2·19-s − 2·31-s − 2·49-s − 2·61-s − 64-s + 2·76-s + 2·79-s + 2·109-s + 2·121-s − 2·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s − 2·196-s + ⋯
L(s)  = 1  + 4-s + 2·19-s − 2·31-s − 2·49-s − 2·61-s − 64-s + 2·76-s + 2·79-s + 2·109-s + 2·121-s − 2·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s − 2·196-s + ⋯

Functional equation

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

Invariants

Degree: \(4\)
Conductor: \(455625\)    =    \(3^{6} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(0.113480\)
Root analytic conductor: \(0.580404\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 455625,\ (\ :0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.078107592\)
\(L(\frac12)\) \(\approx\) \(1.078107592\)
\(L(1)\) 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 \( 1 \)
5 \( 1 \)
good2$C_2^2$ \( 1 - T^{2} + T^{4} \)
7$C_2$ \( ( 1 + T^{2} )^{2} \)
11$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
13$C_2$ \( ( 1 + T^{2} )^{2} \)
17$C_2^2$ \( 1 - T^{2} + T^{4} \)
19$C_2$ \( ( 1 - T + T^{2} )^{2} \)
23$C_2^2$ \( 1 - T^{2} + T^{4} \)
29$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
31$C_2$ \( ( 1 + T + T^{2} )^{2} \)
37$C_2$ \( ( 1 + T^{2} )^{2} \)
41$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
43$C_2$ \( ( 1 + T^{2} )^{2} \)
47$C_2$ \( ( 1 + T^{2} )^{2} \)
53$C_2^2$ \( 1 - T^{2} + T^{4} \)
59$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
61$C_2$ \( ( 1 + T + T^{2} )^{2} \)
67$C_2$ \( ( 1 + T^{2} )^{2} \)
71$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
73$C_2$ \( ( 1 + T^{2} )^{2} \)
79$C_2$ \( ( 1 - T + T^{2} )^{2} \)
83$C_2^2$ \( 1 - T^{2} + T^{4} \)
89$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
97$C_2$ \( ( 1 + T^{2} )^{2} \)
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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

−10.90492307991078464450421584878, −10.69995353470028601555407630088, −10.04233191321693192879368578912, −9.585978252600445084794369590346, −9.293931448809770347876976610029, −8.902673805224168261443791993009, −8.187685716621138649985745615138, −7.77029512094824187863475239894, −7.34647399746674689781816656542, −7.10421018651269618562981977934, −6.54187402819094240718011622585, −5.96035021983221161950886139058, −5.70846878066531994409813910002, −4.82518356472865274244022063026, −4.81568163406284992066207298275, −3.58512094867504956576191343752, −3.44220510745738028070359012889, −2.73689939642412253442071093737, −1.99624806850556960155525512998, −1.38214971959858102173434535003, 1.38214971959858102173434535003, 1.99624806850556960155525512998, 2.73689939642412253442071093737, 3.44220510745738028070359012889, 3.58512094867504956576191343752, 4.81568163406284992066207298275, 4.82518356472865274244022063026, 5.70846878066531994409813910002, 5.96035021983221161950886139058, 6.54187402819094240718011622585, 7.10421018651269618562981977934, 7.34647399746674689781816656542, 7.77029512094824187863475239894, 8.187685716621138649985745615138, 8.902673805224168261443791993009, 9.293931448809770347876976610029, 9.585978252600445084794369590346, 10.04233191321693192879368578912, 10.69995353470028601555407630088, 10.90492307991078464450421584878

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