Properties

Label 4-900081-1.1-c1e2-0-7
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 2·4-s + 2·7-s − 2·9-s + 2·12-s + 3·13-s − 11·19-s + 2·21-s + 2·25-s − 5·27-s + 4·28-s − 8·31-s − 4·36-s − 5·37-s + 3·39-s − 5·43-s + 3·49-s + 6·52-s − 11·57-s + 7·61-s − 4·63-s − 8·64-s − 17·67-s − 8·73-s + 2·75-s − 22·76-s + 79-s + ⋯
L(s)  = 1  + 0.577·3-s + 4-s + 0.755·7-s − 2/3·9-s + 0.577·12-s + 0.832·13-s − 2.52·19-s + 0.436·21-s + 2/5·25-s − 0.962·27-s + 0.755·28-s − 1.43·31-s − 2/3·36-s − 0.821·37-s + 0.480·39-s − 0.762·43-s + 3/7·49-s + 0.832·52-s − 1.45·57-s + 0.896·61-s − 0.503·63-s − 64-s − 2.07·67-s − 0.936·73-s + 0.230·75-s − 2.52·76-s + 0.112·79-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: \(1\)
Selberg data: \((4,\ 900081,\ (\ :1/2, 1/2),\ -1)\)

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$C_2$ \( 1 - T + p T^{2} \)
7$C_1$ \( ( 1 - T )^{2} \)
13$C_1$$\times$$C_2$ \( ( 1 - T )( 1 - 2 T + p T^{2} ) \)
157$C_1$$\times$$C_2$ \( ( 1 - T )( 1 - 2 T + p T^{2} ) \)
good2$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
5$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 - 11 T^{2} + p^{2} T^{4} \)
19$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
23$C_2^2$ \( 1 - 8 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
31$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
37$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \)
43$C_2$$\times$$C_2$ \( ( 1 + T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
47$C_2^2$ \( 1 - 32 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 88 T^{2} + p^{2} T^{4} \)
59$C_2^2$ \( 1 + 22 T^{2} + p^{2} T^{4} \)
61$C_2$$\times$$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 - 2 T + p T^{2} ) \)
67$C_2$$\times$$C_2$ \( ( 1 + 7 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
71$C_2^2$ \( 1 + 40 T^{2} + p^{2} T^{4} \)
73$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
79$C_2$$\times$$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
83$C_2^2$ \( 1 + 133 T^{2} + p^{2} T^{4} \)
89$C_2^2$ \( 1 + 46 T^{2} + p^{2} T^{4} \)
97$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
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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

−8.110479301453071924515405109482, −7.41021947050520686308459675003, −7.22330279339586710753441388576, −6.50725702882899774283565452070, −6.34764257128806342966548928161, −5.79164856060274358157546433179, −5.32781672695916574842398533759, −4.70014575299081980867156491943, −4.13428408432570462598088570484, −3.71133760876523543725615105939, −3.03793408208876605735778958360, −2.49484022402714393518659067190, −1.92020055369410779639093020367, −1.57247026758016037799906478461, 0, 1.57247026758016037799906478461, 1.92020055369410779639093020367, 2.49484022402714393518659067190, 3.03793408208876605735778958360, 3.71133760876523543725615105939, 4.13428408432570462598088570484, 4.70014575299081980867156491943, 5.32781672695916574842398533759, 5.79164856060274358157546433179, 6.34764257128806342966548928161, 6.50725702882899774283565452070, 7.22330279339586710753441388576, 7.41021947050520686308459675003, 8.110479301453071924515405109482

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