Properties

Label 4-2352e2-1.1-c1e2-0-34
Degree $4$
Conductor $5531904$
Sign $1$
Analytic cond. $352.718$
Root an. cond. $4.33368$
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  + 2·3-s − 4·5-s + 3·9-s − 4·11-s − 8·15-s + 4·17-s + 8·19-s − 4·23-s + 4·25-s + 4·27-s + 16·31-s − 8·33-s − 8·37-s + 4·41-s + 16·43-s − 12·45-s + 8·47-s + 8·51-s − 4·53-s + 16·55-s + 16·57-s + 16·59-s − 8·61-s + 16·67-s − 8·69-s − 4·71-s + 8·73-s + ⋯
L(s)  = 1  + 1.15·3-s − 1.78·5-s + 9-s − 1.20·11-s − 2.06·15-s + 0.970·17-s + 1.83·19-s − 0.834·23-s + 4/5·25-s + 0.769·27-s + 2.87·31-s − 1.39·33-s − 1.31·37-s + 0.624·41-s + 2.43·43-s − 1.78·45-s + 1.16·47-s + 1.12·51-s − 0.549·53-s + 2.15·55-s + 2.11·57-s + 2.08·59-s − 1.02·61-s + 1.95·67-s − 0.963·69-s − 0.474·71-s + 0.936·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5531904 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5531904 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(5531904\)    =    \(2^{8} \cdot 3^{2} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(352.718\)
Root analytic conductor: \(4.33368\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 5531904,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.047588157\)
\(L(\frac12)\) \(\approx\) \(3.047588157\)
\(L(\frac{3}{2})\) 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)$
bad2 \( 1 \)
3$C_1$ \( ( 1 - T )^{2} \)
7 \( 1 \)
good5$D_{4}$ \( 1 + 4 T + 12 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
13$C_2^2$ \( 1 + 24 T^{2} + p^{2} T^{4} \)
17$D_{4}$ \( 1 - 4 T + 20 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 - 8 T + 46 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
23$C_4$ \( 1 + 4 T + 42 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \)
31$D_{4}$ \( 1 - 16 T + 118 T^{2} - 16 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 + 8 T + 58 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 - 4 T + 84 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
43$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
47$D_{4}$ \( 1 - 8 T + 102 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 4 T - 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 - 16 T + 174 T^{2} - 16 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 8 T + 40 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
67$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
71$D_{4}$ \( 1 + 4 T + 138 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 8 T + 112 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 - 16 T + 190 T^{2} - 16 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - 8 T + 54 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 4 T + 20 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 24 T + 320 T^{2} - 24 p T^{3} + p^{2} T^{4} \)
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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

−9.051833350969874694704352360075, −8.708158176051700115869362542776, −8.117807387412068599463137070272, −8.021337914721385923227782978657, −7.75490448730835657912510225579, −7.52597829640397611346636251485, −7.14775031270455405022195628679, −6.65742998796066199752008478439, −5.98732119808227100832063966567, −5.67248094492287799687207376224, −4.95533941277000983094691603980, −4.86772632215215173156181422525, −4.05938233261773429115816266020, −3.96049629492318870374659752523, −3.27871121143100104399319258404, −3.22841826178006948893098390296, −2.41649035239950971977131476626, −2.25309203672834778486684851137, −0.975969310720986727077965539905, −0.71359646878133320903381307532, 0.71359646878133320903381307532, 0.975969310720986727077965539905, 2.25309203672834778486684851137, 2.41649035239950971977131476626, 3.22841826178006948893098390296, 3.27871121143100104399319258404, 3.96049629492318870374659752523, 4.05938233261773429115816266020, 4.86772632215215173156181422525, 4.95533941277000983094691603980, 5.67248094492287799687207376224, 5.98732119808227100832063966567, 6.65742998796066199752008478439, 7.14775031270455405022195628679, 7.52597829640397611346636251485, 7.75490448730835657912510225579, 8.021337914721385923227782978657, 8.117807387412068599463137070272, 8.708158176051700115869362542776, 9.051833350969874694704352360075

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