Properties

Label 4-294e2-1.1-c3e2-0-9
Degree $4$
Conductor $86436$
Sign $1$
Analytic cond. $300.903$
Root an. cond. $4.16492$
Motivic weight $3$
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  + 2·2-s − 3·3-s − 15·5-s − 6·6-s − 8·8-s − 30·10-s + 9·11-s + 176·13-s + 45·15-s − 16·16-s − 84·17-s + 104·19-s + 18·22-s + 84·23-s + 24·24-s + 125·25-s + 352·26-s + 27·27-s + 102·29-s + 90·30-s + 185·31-s − 27·33-s − 168·34-s − 44·37-s + 208·38-s − 528·39-s + 120·40-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s − 1.34·5-s − 0.408·6-s − 0.353·8-s − 0.948·10-s + 0.246·11-s + 3.75·13-s + 0.774·15-s − 1/4·16-s − 1.19·17-s + 1.25·19-s + 0.174·22-s + 0.761·23-s + 0.204·24-s + 25-s + 2.65·26-s + 0.192·27-s + 0.653·29-s + 0.547·30-s + 1.07·31-s − 0.142·33-s − 0.847·34-s − 0.195·37-s + 0.887·38-s − 2.16·39-s + 0.474·40-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 86436 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 86436 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(86436\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(300.903\)
Root analytic conductor: \(4.16492\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{294} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 86436,\ (\ :3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(3.065169152\)
\(L(\frac12)\) \(\approx\) \(3.065169152\)
\(L(\frac{5}{2})\) 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$C_2$ \( 1 - p T + p^{2} T^{2} \)
3$C_2$ \( 1 + p T + p^{2} T^{2} \)
7 \( 1 \)
good5$C_2^2$ \( 1 + 3 p T + 4 p^{2} T^{2} + 3 p^{4} T^{3} + p^{6} T^{4} \)
11$C_2^2$ \( 1 - 9 T - 1250 T^{2} - 9 p^{3} T^{3} + p^{6} T^{4} \)
13$C_2$ \( ( 1 - 88 T + p^{3} T^{2} )^{2} \)
17$C_2^2$ \( 1 + 84 T + 2143 T^{2} + 84 p^{3} T^{3} + p^{6} T^{4} \)
19$C_2^2$ \( 1 - 104 T + 3957 T^{2} - 104 p^{3} T^{3} + p^{6} T^{4} \)
23$C_2^2$ \( 1 - 84 T - 5111 T^{2} - 84 p^{3} T^{3} + p^{6} T^{4} \)
29$C_2$ \( ( 1 - 51 T + p^{3} T^{2} )^{2} \)
31$C_2^2$ \( 1 - 185 T + 4434 T^{2} - 185 p^{3} T^{3} + p^{6} T^{4} \)
37$C_2^2$ \( 1 + 44 T - 48717 T^{2} + 44 p^{3} T^{3} + p^{6} T^{4} \)
41$C_2$ \( ( 1 - 168 T + p^{3} T^{2} )^{2} \)
43$C_2$ \( ( 1 - 326 T + p^{3} T^{2} )^{2} \)
47$C_2^2$ \( 1 + 138 T - 84779 T^{2} + 138 p^{3} T^{3} + p^{6} T^{4} \)
53$C_2^2$ \( 1 + 639 T + 259444 T^{2} + 639 p^{3} T^{3} + p^{6} T^{4} \)
59$C_2^2$ \( 1 - 159 T - 180098 T^{2} - 159 p^{3} T^{3} + p^{6} T^{4} \)
61$C_2^2$ \( 1 - 722 T + 294303 T^{2} - 722 p^{3} T^{3} + p^{6} T^{4} \)
67$C_2^2$ \( 1 - 166 T - 273207 T^{2} - 166 p^{3} T^{3} + p^{6} T^{4} \)
71$C_2$ \( ( 1 - 1086 T + p^{3} T^{2} )^{2} \)
73$C_2^2$ \( 1 - 218 T - 341493 T^{2} - 218 p^{3} T^{3} + p^{6} T^{4} \)
79$C_2^2$ \( 1 - 583 T - 153150 T^{2} - 583 p^{3} T^{3} + p^{6} T^{4} \)
83$C_2$ \( ( 1 - 597 T + p^{3} T^{2} )^{2} \)
89$C_2^2$ \( 1 + 1038 T + 372475 T^{2} + 1038 p^{3} T^{3} + p^{6} T^{4} \)
97$C_2$ \( ( 1 - 169 T + p^{3} T^{2} )^{2} \)
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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

−11.59167692088573204120501530210, −11.06623633803055245897894297780, −10.86052495199959519896007248858, −10.75211630743941007668891537802, −9.383109024441158913985252014042, −9.335125065489385948464744336181, −8.530715602160097581742047109181, −8.260126409059750946534249849781, −7.87490243641093422137524217029, −6.97368471538113584795911811639, −6.41390495457370764148171256329, −6.27720941470193590136274175373, −5.54849671406903208552627925034, −4.93006153973490044115614601044, −4.26795454360335452991228279177, −3.67990035347657234467470825440, −3.59540763886032649512861377942, −2.60408323786689834479359654183, −1.04725889721364366007117534790, −0.830477582989188148630957599919, 0.830477582989188148630957599919, 1.04725889721364366007117534790, 2.60408323786689834479359654183, 3.59540763886032649512861377942, 3.67990035347657234467470825440, 4.26795454360335452991228279177, 4.93006153973490044115614601044, 5.54849671406903208552627925034, 6.27720941470193590136274175373, 6.41390495457370764148171256329, 6.97368471538113584795911811639, 7.87490243641093422137524217029, 8.260126409059750946534249849781, 8.530715602160097581742047109181, 9.335125065489385948464744336181, 9.383109024441158913985252014042, 10.75211630743941007668891537802, 10.86052495199959519896007248858, 11.06623633803055245897894297780, 11.59167692088573204120501530210

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