Properties

Label 4-147e2-1.1-c3e2-0-9
Degree $4$
Conductor $21609$
Sign $1$
Analytic cond. $75.2257$
Root an. cond. $2.94504$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + 3·2-s + 3·3-s + 8·4-s − 3·5-s + 9·6-s + 45·8-s − 9·10-s + 15·11-s + 24·12-s + 128·13-s − 9·15-s + 135·16-s + 84·17-s − 16·19-s − 24·20-s + 45·22-s + 84·23-s + 135·24-s + 125·25-s + 384·26-s − 27·27-s − 594·29-s − 27·30-s − 253·31-s + 360·32-s + 45·33-s + 252·34-s + ⋯
L(s)  = 1  + 1.06·2-s + 0.577·3-s + 4-s − 0.268·5-s + 0.612·6-s + 1.98·8-s − 0.284·10-s + 0.411·11-s + 0.577·12-s + 2.73·13-s − 0.154·15-s + 2.10·16-s + 1.19·17-s − 0.193·19-s − 0.268·20-s + 0.436·22-s + 0.761·23-s + 1.14·24-s + 25-s + 2.89·26-s − 0.192·27-s − 3.80·29-s − 0.164·30-s − 1.46·31-s + 1.98·32-s + 0.237·33-s + 1.27·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 21609 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21609 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(21609\)    =    \(3^{2} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(75.2257\)
Root analytic conductor: \(2.94504\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{147} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 21609,\ (\ :3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(7.071238859\)
\(L(\frac12)\) \(\approx\) \(7.071238859\)
\(L(\frac{5}{2})\) 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)$
bad3$C_2$ \( 1 - p T + p^{2} T^{2} \)
7 \( 1 \)
good2$C_2^2$ \( 1 - 3 T + T^{2} - 3 p^{3} T^{3} + p^{6} T^{4} \)
5$C_2^2$ \( 1 + 3 T - 116 T^{2} + 3 p^{3} T^{3} + p^{6} T^{4} \)
11$C_2^2$ \( 1 - 15 T - 1106 T^{2} - 15 p^{3} T^{3} + p^{6} T^{4} \)
13$C_2$ \( ( 1 - 64 T + p^{3} T^{2} )^{2} \)
17$C_2^2$ \( 1 - 84 T + 2143 T^{2} - 84 p^{3} T^{3} + p^{6} T^{4} \)
19$C_2^2$ \( 1 + 16 T - 6603 T^{2} + 16 p^{3} T^{3} + p^{6} T^{4} \)
23$C_2^2$ \( 1 - 84 T - 5111 T^{2} - 84 p^{3} T^{3} + p^{6} T^{4} \)
29$C_2$ \( ( 1 + 297 T + p^{3} T^{2} )^{2} \)
31$C_2^2$ \( 1 + 253 T + 34218 T^{2} + 253 p^{3} T^{3} + p^{6} T^{4} \)
37$C_2^2$ \( 1 - 316 T + 49203 T^{2} - 316 p^{3} T^{3} + p^{6} T^{4} \)
41$C_2$ \( ( 1 + 360 T + p^{3} T^{2} )^{2} \)
43$C_2$ \( ( 1 - 26 T + p^{3} T^{2} )^{2} \)
47$C_2^2$ \( 1 + 30 T - 102923 T^{2} + 30 p^{3} T^{3} + p^{6} T^{4} \)
53$C_2^2$ \( 1 + 363 T - 17108 T^{2} + 363 p^{3} T^{3} + p^{6} T^{4} \)
59$C_2^2$ \( 1 + 15 T - 205154 T^{2} + 15 p^{3} T^{3} + p^{6} T^{4} \)
61$C_2^2$ \( 1 + 118 T - 213057 T^{2} + 118 p^{3} T^{3} + p^{6} T^{4} \)
67$C_2^2$ \( 1 - 370 T - 163863 T^{2} - 370 p^{3} T^{3} + p^{6} T^{4} \)
71$C_2$ \( ( 1 + 342 T + p^{3} T^{2} )^{2} \)
73$C_2^2$ \( 1 - 362 T - 257973 T^{2} - 362 p^{3} T^{3} + p^{6} T^{4} \)
79$C_2^2$ \( 1 + 467 T - 274950 T^{2} + 467 p^{3} T^{3} + p^{6} T^{4} \)
83$C_2$ \( ( 1 + 477 T + p^{3} T^{2} )^{2} \)
89$C_2^2$ \( 1 - 906 T + 115867 T^{2} - 906 p^{3} T^{3} + p^{6} T^{4} \)
97$C_2$ \( ( 1 + 503 T + p^{3} T^{2} )^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−13.05253585052028495242296659509, −12.70896822669867857549390891060, −11.71474196055688402396418224015, −11.31591211882210635298441118862, −10.89723522390686879738388645702, −10.72163773940034770007008903459, −9.764356135350725708438517771339, −9.210009467697763846336108466812, −8.505395791247484752971236149756, −8.131431321827146537120156053627, −7.33475489140444438216219836936, −7.12422991627852628487606547170, −6.14557377251303475641347644439, −5.70546428097437895329684621541, −5.06508962445991743858441987979, −4.10924363295733866121086820539, −3.49140915880193765976326389736, −3.40932366894813089321204557860, −1.78412743762629322276598970783, −1.32128003035094388887531434919, 1.32128003035094388887531434919, 1.78412743762629322276598970783, 3.40932366894813089321204557860, 3.49140915880193765976326389736, 4.10924363295733866121086820539, 5.06508962445991743858441987979, 5.70546428097437895329684621541, 6.14557377251303475641347644439, 7.12422991627852628487606547170, 7.33475489140444438216219836936, 8.131431321827146537120156053627, 8.505395791247484752971236149756, 9.210009467697763846336108466812, 9.764356135350725708438517771339, 10.72163773940034770007008903459, 10.89723522390686879738388645702, 11.31591211882210635298441118862, 11.71474196055688402396418224015, 12.70896822669867857549390891060, 13.05253585052028495242296659509

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