Properties

Label 4-18e2-1.1-c3e2-0-0
Degree $4$
Conductor $324$
Sign $1$
Analytic cond. $1.12791$
Root an. cond. $1.03055$
Motivic weight $3$
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  − 2·2-s + 9·5-s + 31·7-s + 8·8-s − 27·9-s − 18·10-s + 15·11-s + 37·13-s − 62·14-s − 16·16-s − 84·17-s + 54·18-s − 56·19-s − 30·22-s − 195·23-s + 125·25-s − 74·26-s − 111·29-s + 205·31-s + 168·34-s + 279·35-s − 332·37-s + 112·38-s + 72·40-s + 261·41-s + 43·43-s − 243·45-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.804·5-s + 1.67·7-s + 0.353·8-s − 9-s − 0.569·10-s + 0.411·11-s + 0.789·13-s − 1.18·14-s − 1/4·16-s − 1.19·17-s + 0.707·18-s − 0.676·19-s − 0.290·22-s − 1.76·23-s + 25-s − 0.558·26-s − 0.710·29-s + 1.18·31-s + 0.847·34-s + 1.34·35-s − 1.47·37-s + 0.478·38-s + 0.284·40-s + 0.994·41-s + 0.152·43-s − 0.804·45-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(324\)    =    \(2^{2} \cdot 3^{4}\)
Sign: $1$
Analytic conductor: \(1.12791\)
Root analytic conductor: \(1.03055\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 324,\ (\ :3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(0.8863012641\)
\(L(\frac12)\) \(\approx\) \(0.8863012641\)
\(L(\frac{5}{2})\) 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$C_2$ \( 1 + p^{3} T^{2} \)
good5$C_2^2$ \( 1 - 9 T - 44 T^{2} - 9 p^{3} T^{3} + p^{6} T^{4} \)
7$C_2^2$ \( 1 - 31 T + 618 T^{2} - 31 p^{3} T^{3} + p^{6} T^{4} \)
11$C_2^2$ \( 1 - 15 T - 1106 T^{2} - 15 p^{3} T^{3} + p^{6} T^{4} \)
13$C_2^2$ \( 1 - 37 T - 828 T^{2} - 37 p^{3} T^{3} + p^{6} T^{4} \)
17$C_2$ \( ( 1 + 42 T + p^{3} T^{2} )^{2} \)
19$C_2$ \( ( 1 + 28 T + p^{3} T^{2} )^{2} \)
23$C_2^2$ \( 1 + 195 T + 25858 T^{2} + 195 p^{3} T^{3} + p^{6} T^{4} \)
29$C_2^2$ \( 1 + 111 T - 12068 T^{2} + 111 p^{3} T^{3} + p^{6} T^{4} \)
31$C_2^2$ \( 1 - 205 T + 12234 T^{2} - 205 p^{3} T^{3} + p^{6} T^{4} \)
37$C_2$ \( ( 1 + 166 T + p^{3} T^{2} )^{2} \)
41$C_2^2$ \( 1 - 261 T - 800 T^{2} - 261 p^{3} T^{3} + p^{6} T^{4} \)
43$C_2^2$ \( 1 - p T - 42 p^{2} T^{2} - p^{4} T^{3} + p^{6} T^{4} \)
47$C_2^2$ \( 1 + 177 T - 72494 T^{2} + 177 p^{3} T^{3} + p^{6} T^{4} \)
53$C_2$ \( ( 1 - 114 T + p^{3} T^{2} )^{2} \)
59$C_2^2$ \( 1 + 159 T - 180098 T^{2} + 159 p^{3} T^{3} + p^{6} T^{4} \)
61$C_2^2$ \( 1 + 191 T - 190500 T^{2} + 191 p^{3} T^{3} + p^{6} T^{4} \)
67$C_2^2$ \( 1 - 421 T - 123522 T^{2} - 421 p^{3} T^{3} + p^{6} T^{4} \)
71$C_2$ \( ( 1 - 156 T + p^{3} T^{2} )^{2} \)
73$C_2$ \( ( 1 - 182 T + p^{3} T^{2} )^{2} \)
79$C_2^2$ \( 1 + 1133 T + 790650 T^{2} + 1133 p^{3} T^{3} + p^{6} T^{4} \)
83$C_2^2$ \( 1 - 1083 T + 601102 T^{2} - 1083 p^{3} T^{3} + p^{6} T^{4} \)
89$C_2$ \( ( 1 + 1050 T + p^{3} T^{2} )^{2} \)
97$C_2^2$ \( 1 - 901 T - 100872 T^{2} - 901 p^{3} T^{3} + p^{6} T^{4} \)
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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

−18.14228405724161650303447802579, −17.98010892956469024606134947500, −17.32899427275944951131865274870, −17.07765926648066177146356404859, −16.13157874639066754573935098721, −15.29020789404722998881645073590, −14.48447618557902344058107544074, −14.00263829233119909279857834873, −13.54289523139430969864649595321, −12.40302537557065707422760620618, −11.42938516121310386509632186967, −11.03224542513244268239564416890, −10.25633008352779349364677513132, −9.168554738670215362145146658577, −8.509455018666476487755178860952, −8.062690670972476804372231461517, −6.60179753450900366957381205288, −5.60789669161822730948025485554, −4.38699503553042529087882060291, −1.94503011243440762792153454089, 1.94503011243440762792153454089, 4.38699503553042529087882060291, 5.60789669161822730948025485554, 6.60179753450900366957381205288, 8.062690670972476804372231461517, 8.509455018666476487755178860952, 9.168554738670215362145146658577, 10.25633008352779349364677513132, 11.03224542513244268239564416890, 11.42938516121310386509632186967, 12.40302537557065707422760620618, 13.54289523139430969864649595321, 14.00263829233119909279857834873, 14.48447618557902344058107544074, 15.29020789404722998881645073590, 16.13157874639066754573935098721, 17.07765926648066177146356404859, 17.32899427275944951131865274870, 17.98010892956469024606134947500, 18.14228405724161650303447802579

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