Properties

Label 4-6864e2-1.1-c1e2-0-1
Degree $4$
Conductor $47114496$
Sign $1$
Analytic cond. $3004.06$
Root an. cond. $7.40333$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s + 5-s + 7-s + 3·9-s + 2·11-s − 2·13-s + 2·15-s − 8·17-s + 2·19-s + 2·21-s + 23-s − 25-s + 4·27-s − 29-s − 6·31-s + 4·33-s + 35-s − 4·37-s − 4·39-s + 5·41-s + 11·43-s + 3·45-s + 8·47-s − 5·49-s − 16·51-s − 20·53-s + 2·55-s + ⋯
L(s)  = 1  + 1.15·3-s + 0.447·5-s + 0.377·7-s + 9-s + 0.603·11-s − 0.554·13-s + 0.516·15-s − 1.94·17-s + 0.458·19-s + 0.436·21-s + 0.208·23-s − 1/5·25-s + 0.769·27-s − 0.185·29-s − 1.07·31-s + 0.696·33-s + 0.169·35-s − 0.657·37-s − 0.640·39-s + 0.780·41-s + 1.67·43-s + 0.447·45-s + 1.16·47-s − 5/7·49-s − 2.24·51-s − 2.74·53-s + 0.269·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 47114496 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 47114496 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(47114496\)    =    \(2^{8} \cdot 3^{2} \cdot 11^{2} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(3004.06\)
Root analytic conductor: \(7.40333\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 47114496,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.886318117\)
\(L(\frac12)\) \(\approx\) \(4.886318117\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{2} \)
11$C_1$ \( ( 1 - T )^{2} \)
13$C_1$ \( ( 1 + T )^{2} \)
good5$D_{4}$ \( 1 - T + 2 T^{2} - p T^{3} + p^{2} T^{4} \)
7$D_{4}$ \( 1 - T + 6 T^{2} - p T^{3} + p^{2} T^{4} \)
17$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
19$D_{4}$ \( 1 - 2 T + 6 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 - T + 38 T^{2} - p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 + T + 50 T^{2} + p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 + 6 T + 38 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
37$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
41$D_{4}$ \( 1 - 5 T + 80 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 - 11 T + 108 T^{2} - 11 p T^{3} + p^{2} T^{4} \)
47$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
53$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
59$D_{4}$ \( 1 - 17 T + 182 T^{2} - 17 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + T + 48 T^{2} + p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + 7 T + 72 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 - 18 T + 190 T^{2} - 18 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - T + 72 T^{2} - p T^{3} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
83$D_{4}$ \( 1 - 6 T + 142 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
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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

−8.058508586655017691453983713457, −7.925420226412624068642243432977, −7.31138508710785040308044885915, −7.30973427211383251116458737653, −6.66154573941343039170716073729, −6.60289333364216568033246011246, −6.13836175738851683204064811013, −5.62494640243060875777466899097, −5.26481254065914092328052368834, −4.87070638132792079920474002832, −4.48660913779082125170326539439, −4.17025552249705982111939195818, −3.67569748612377254821245771520, −3.50057662720215514638876296515, −2.80438771709272235143884966835, −2.46786281578398541566464297111, −1.99979098434646542758184272738, −1.89198308038529015901832397676, −1.15293157346452956141989539434, −0.49852570388615041742254469647, 0.49852570388615041742254469647, 1.15293157346452956141989539434, 1.89198308038529015901832397676, 1.99979098434646542758184272738, 2.46786281578398541566464297111, 2.80438771709272235143884966835, 3.50057662720215514638876296515, 3.67569748612377254821245771520, 4.17025552249705982111939195818, 4.48660913779082125170326539439, 4.87070638132792079920474002832, 5.26481254065914092328052368834, 5.62494640243060875777466899097, 6.13836175738851683204064811013, 6.60289333364216568033246011246, 6.66154573941343039170716073729, 7.30973427211383251116458737653, 7.31138508710785040308044885915, 7.925420226412624068642243432977, 8.058508586655017691453983713457

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