Properties

Label 4-448e2-1.1-c1e2-0-8
Degree $4$
Conductor $200704$
Sign $1$
Analytic cond. $12.7970$
Root an. cond. $1.89137$
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 − 2·5-s + 2·7-s + 2·9-s + 4·11-s − 6·13-s − 4·15-s − 2·19-s + 4·21-s + 8·23-s − 2·25-s + 6·27-s + 4·31-s + 8·33-s − 4·35-s − 12·39-s − 8·41-s − 4·43-s − 4·45-s + 12·47-s + 3·49-s + 20·53-s − 8·55-s − 4·57-s + 14·59-s − 18·61-s + 4·63-s + ⋯
L(s)  = 1  + 1.15·3-s − 0.894·5-s + 0.755·7-s + 2/3·9-s + 1.20·11-s − 1.66·13-s − 1.03·15-s − 0.458·19-s + 0.872·21-s + 1.66·23-s − 2/5·25-s + 1.15·27-s + 0.718·31-s + 1.39·33-s − 0.676·35-s − 1.92·39-s − 1.24·41-s − 0.609·43-s − 0.596·45-s + 1.75·47-s + 3/7·49-s + 2.74·53-s − 1.07·55-s − 0.529·57-s + 1.82·59-s − 2.30·61-s + 0.503·63-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 200704 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 200704 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(200704\)    =    \(2^{12} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(12.7970\)
Root analytic conductor: \(1.89137\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 200704,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.397089755\)
\(L(\frac12)\) \(\approx\) \(2.397089755\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7$C_1$ \( ( 1 - T )^{2} \)
good3$C_2^2$ \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
5$D_{4}$ \( 1 + 2 T + 6 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
11$C_4$ \( 1 - 4 T + 6 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 + 6 T + 30 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
17$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
19$D_{4}$ \( 1 + 2 T + 34 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
23$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
29$C_2^2$ \( 1 + 38 T^{2} + p^{2} T^{4} \)
31$C_4$ \( 1 - 4 T + 46 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
37$C_2^2$ \( 1 + 54 T^{2} + p^{2} T^{4} \)
41$D_{4}$ \( 1 + 8 T + 78 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 + 4 T + 70 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 - 12 T + 110 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
53$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
59$D_{4}$ \( 1 - 14 T + 162 T^{2} - 14 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 18 T + 198 T^{2} + 18 p T^{3} + p^{2} T^{4} \)
67$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
71$D_{4}$ \( 1 - 8 T + 78 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 12 T + 102 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 - 8 T + 94 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - 14 T + 210 T^{2} - 14 p T^{3} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
97$D_{4}$ \( 1 - 16 T + 238 T^{2} - 16 p T^{3} + p^{2} T^{4} \)
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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

−11.40447279000508985598442576999, −10.78791531696252790432431091309, −10.46069723831881092012825615668, −9.928580108846962340176461681333, −9.241643254315026546593985324918, −9.154303778656024394519853196568, −8.425631430175080871150353911804, −8.366086263278235606408736912743, −7.62607127682092283007091123577, −7.35129617628906922001146181175, −6.80097529692576066096871324143, −6.51146798637493478259712053812, −5.31296608222213756198279977427, −5.10693491524925785420875481967, −4.24185804371242371066282270128, −4.10132448669145396627282680616, −3.29794345337559195465054323068, −2.65317914822623940275315184375, −2.07760455765410581888397145808, −0.976012793566189189926165323570, 0.976012793566189189926165323570, 2.07760455765410581888397145808, 2.65317914822623940275315184375, 3.29794345337559195465054323068, 4.10132448669145396627282680616, 4.24185804371242371066282270128, 5.10693491524925785420875481967, 5.31296608222213756198279977427, 6.51146798637493478259712053812, 6.80097529692576066096871324143, 7.35129617628906922001146181175, 7.62607127682092283007091123577, 8.366086263278235606408736912743, 8.425631430175080871150353911804, 9.154303778656024394519853196568, 9.241643254315026546593985324918, 9.928580108846962340176461681333, 10.46069723831881092012825615668, 10.78791531696252790432431091309, 11.40447279000508985598442576999

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