Properties

Label 4-900081-1.1-c1e2-0-3
Degree $4$
Conductor $900081$
Sign $1$
Analytic cond. $57.3899$
Root an. cond. $2.75238$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s + 4-s + 3·7-s + 9-s + 2·12-s − 3·13-s − 3·16-s − 19-s + 6·21-s + 4·25-s − 4·27-s + 3·28-s + 4·31-s + 36-s + 15·37-s − 6·39-s + 15·43-s − 6·48-s + 2·49-s − 3·52-s − 2·57-s − 10·61-s + 3·63-s − 7·64-s + 2·67-s − 15·73-s + 8·75-s + ⋯
L(s)  = 1  + 1.15·3-s + 1/2·4-s + 1.13·7-s + 1/3·9-s + 0.577·12-s − 0.832·13-s − 3/4·16-s − 0.229·19-s + 1.30·21-s + 4/5·25-s − 0.769·27-s + 0.566·28-s + 0.718·31-s + 1/6·36-s + 2.46·37-s − 0.960·39-s + 2.28·43-s − 0.866·48-s + 2/7·49-s − 0.416·52-s − 0.264·57-s − 1.28·61-s + 0.377·63-s − 7/8·64-s + 0.244·67-s − 1.75·73-s + 0.923·75-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 900081 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900081 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(900081\)    =    \(3^{2} \cdot 7^{2} \cdot 13 \cdot 157\)
Sign: $1$
Analytic conductor: \(57.3899\)
Root analytic conductor: \(2.75238\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{900081} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 900081,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.940221203\)
\(L(\frac12)\) \(\approx\) \(3.940221203\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - 2 T + p T^{2} \)
7$C_2$ \( 1 - 3 T + p T^{2} \)
13$C_1$$\times$$C_2$ \( ( 1 - T )( 1 + 4 T + p T^{2} ) \)
157$C_1$$\times$$C_2$ \( ( 1 - T )( 1 - 16 T + p T^{2} ) \)
good2$C_2^2$ \( 1 - T^{2} + p^{2} T^{4} \)
5$C_2^2$ \( 1 - 4 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
17$C_2^2$ \( 1 - 4 T^{2} + p^{2} T^{4} \)
19$C_2$$\times$$C_2$ \( ( 1 - T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
23$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + 20 T^{2} + p^{2} T^{4} \)
31$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \)
37$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 - 5 T + p T^{2} ) \)
41$C_2^2$ \( 1 + 16 T^{2} + p^{2} T^{4} \)
43$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 - 7 T + p T^{2} ) \)
47$C_2^2$ \( 1 + 40 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 16 T^{2} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - 73 T^{2} + p^{2} T^{4} \)
61$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
67$C_2$$\times$$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 11 T + p T^{2} ) \)
71$C_2^2$ \( 1 + 63 T^{2} + p^{2} T^{4} \)
73$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 13 T + p T^{2} ) \)
79$C_2$$\times$$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 - 11 T + p T^{2} ) \)
83$C_2^2$ \( 1 - 101 T^{2} + p^{2} T^{4} \)
89$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \)
97$C_2$$\times$$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 - 4 T + p T^{2} ) \)
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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

−8.088601946806000732363349938922, −7.69638747652121587804006211578, −7.52930708222587815122527168550, −7.09356612833747651734026778663, −6.25345201099211424403232306588, −6.19752293639832070781925357609, −5.43271603315015379660814721837, −4.79303604572622621602849104698, −4.48425864080227360653212391988, −4.09392832058938460368797158731, −3.24284307595571860971533479966, −2.66658852520130639214428964159, −2.39610015411637638726645668952, −1.83131597359277800294877006935, −0.881563138032954447033656569676, 0.881563138032954447033656569676, 1.83131597359277800294877006935, 2.39610015411637638726645668952, 2.66658852520130639214428964159, 3.24284307595571860971533479966, 4.09392832058938460368797158731, 4.48425864080227360653212391988, 4.79303604572622621602849104698, 5.43271603315015379660814721837, 6.19752293639832070781925357609, 6.25345201099211424403232306588, 7.09356612833747651734026778663, 7.52930708222587815122527168550, 7.69638747652121587804006211578, 8.088601946806000732363349938922

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