Properties

Label 4-5415e2-1.1-c1e2-0-9
Degree $4$
Conductor $29322225$
Sign $1$
Analytic cond. $1869.61$
Root an. cond. $6.57563$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s − 4-s + 2·5-s − 2·7-s + 3·9-s + 6·11-s + 2·12-s + 2·13-s − 4·15-s − 3·16-s − 2·20-s + 4·21-s + 3·25-s − 4·27-s + 2·28-s − 6·29-s − 4·31-s − 12·33-s − 4·35-s − 3·36-s + 2·37-s − 4·39-s − 6·41-s − 2·43-s − 6·44-s + 6·45-s + 6·48-s + ⋯
L(s)  = 1  − 1.15·3-s − 1/2·4-s + 0.894·5-s − 0.755·7-s + 9-s + 1.80·11-s + 0.577·12-s + 0.554·13-s − 1.03·15-s − 3/4·16-s − 0.447·20-s + 0.872·21-s + 3/5·25-s − 0.769·27-s + 0.377·28-s − 1.11·29-s − 0.718·31-s − 2.08·33-s − 0.676·35-s − 1/2·36-s + 0.328·37-s − 0.640·39-s − 0.937·41-s − 0.304·43-s − 0.904·44-s + 0.894·45-s + 0.866·48-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 29322225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29322225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(29322225\)    =    \(3^{2} \cdot 5^{2} \cdot 19^{4}\)
Sign: $1$
Analytic conductor: \(1869.61\)
Root analytic conductor: \(6.57563\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 29322225,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 + T )^{2} \)
5$C_1$ \( ( 1 - T )^{2} \)
19 \( 1 \)
good2$C_2^2$ \( 1 + T^{2} + p^{2} T^{4} \)
7$D_{4}$ \( 1 + 2 T + 12 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 - 6 T + 28 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 - 2 T + 24 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
17$C_2$ \( ( 1 + p T^{2} )^{2} \)
23$C_2^2$ \( 1 + 34 T^{2} + p^{2} T^{4} \)
29$D_{4}$ \( 1 + 6 T + 40 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 - 2 T + 48 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 + 6 T + 88 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 + 2 T + 60 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
47$C_2^2$ \( 1 + 82 T^{2} + p^{2} T^{4} \)
53$D_{4}$ \( 1 - 12 T + 130 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 12 T + 142 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 20 T + 210 T^{2} + 20 p T^{3} + p^{2} T^{4} \)
67$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
71$D_{4}$ \( 1 + 12 T + 70 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 20 T + 198 T^{2} + 20 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 - 8 T + 126 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + 12 T + 154 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 18 T + 256 T^{2} + 18 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 2 T + 168 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
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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

−7.961071333282791354513617030432, −7.32478401844020461457781769827, −7.22032402910208055767803203691, −6.86499150379127540525290213246, −6.44528184226955045317251295393, −6.04058574568757094916242401717, −5.90051746416864977108158047715, −5.79693992146126944657838424507, −4.91799957204996053220937668842, −4.86827880212800519922536687528, −4.28718030625487614798127875633, −4.05751383096896134034704258424, −3.56684986462496203530113572455, −3.12943770216453842900956451971, −2.65796349454318247575124955266, −1.79933735172390308636248659644, −1.46539775694348188894541597264, −1.26012420311168812593244740108, 0, 0, 1.26012420311168812593244740108, 1.46539775694348188894541597264, 1.79933735172390308636248659644, 2.65796349454318247575124955266, 3.12943770216453842900956451971, 3.56684986462496203530113572455, 4.05751383096896134034704258424, 4.28718030625487614798127875633, 4.86827880212800519922536687528, 4.91799957204996053220937668842, 5.79693992146126944657838424507, 5.90051746416864977108158047715, 6.04058574568757094916242401717, 6.44528184226955045317251295393, 6.86499150379127540525290213246, 7.22032402910208055767803203691, 7.32478401844020461457781769827, 7.961071333282791354513617030432

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