Properties

Degree $4$
Conductor $21609$
Sign $1$
Motivic weight $1$
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 + 2·3-s + 4-s + 4·5-s − 4·6-s + 3·9-s − 8·10-s − 4·11-s + 2·12-s + 8·13-s + 8·15-s + 16-s + 4·17-s − 6·18-s + 4·20-s + 8·22-s − 4·23-s + 4·25-s − 16·26-s + 4·27-s − 8·29-s − 16·30-s − 8·31-s + 2·32-s − 8·33-s − 8·34-s + 3·36-s + ⋯
L(s)  = 1  − 1.41·2-s + 1.15·3-s + 1/2·4-s + 1.78·5-s − 1.63·6-s + 9-s − 2.52·10-s − 1.20·11-s + 0.577·12-s + 2.21·13-s + 2.06·15-s + 1/4·16-s + 0.970·17-s − 1.41·18-s + 0.894·20-s + 1.70·22-s − 0.834·23-s + 4/5·25-s − 3.13·26-s + 0.769·27-s − 1.48·29-s − 2.92·30-s − 1.43·31-s + 0.353·32-s − 1.39·33-s − 1.37·34-s + 1/2·36-s + ⋯

Functional equation

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

Invariants

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

Particular Values

\(L(1)\) \(\approx\) \(1.00228\)
\(L(\frac12)\) \(\approx\) \(1.00228\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{2} \)
7 \( 1 \)
good2$D_{4}$ \( 1 + p T + 3 T^{2} + p^{2} T^{3} + p^{2} T^{4} \)
5$D_{4}$ \( 1 - 4 T + 12 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
11$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
13$D_{4}$ \( 1 - 8 T + 40 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 - 4 T + 20 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
19$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \)
23$D_{4}$ \( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 + 8 T + 66 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 + 8 T + 70 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
37$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
41$D_{4}$ \( 1 - 4 T + 68 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
43$C_2^2$ \( 1 + 54 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 + 86 T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
59$D_{4}$ \( 1 + 8 T + 126 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 - 16 T + 168 T^{2} - 16 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + 102 T^{2} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 8 T + 64 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 - 16 T + 190 T^{2} - 16 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + 8 T + 54 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 20 T + 260 T^{2} + 20 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 8 T + 208 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
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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

−13.27991901401338177920979096245, −13.06252264540380797503225997297, −12.58858414642191942982135369824, −11.64815720039895623060251437167, −10.77716644511339454537220883340, −10.63626188559369251023440765868, −9.827557758340378217934002094453, −9.783092558409999515652215080723, −9.076856079002227489979725189354, −8.913542592088079760457981218470, −8.059283027700205192169067404971, −8.052294528463012176197637921293, −7.19369438058058265519089458762, −6.28194973118784079134047204927, −5.71762903499348270420993315024, −5.35315698187524417229152126700, −3.90425253661183828324360261573, −3.29104047009395636747641337910, −2.16504759383512122024690547881, −1.51610409940336560074145371519, 1.51610409940336560074145371519, 2.16504759383512122024690547881, 3.29104047009395636747641337910, 3.90425253661183828324360261573, 5.35315698187524417229152126700, 5.71762903499348270420993315024, 6.28194973118784079134047204927, 7.19369438058058265519089458762, 8.052294528463012176197637921293, 8.059283027700205192169067404971, 8.913542592088079760457981218470, 9.076856079002227489979725189354, 9.783092558409999515652215080723, 9.827557758340378217934002094453, 10.63626188559369251023440765868, 10.77716644511339454537220883340, 11.64815720039895623060251437167, 12.58858414642191942982135369824, 13.06252264540380797503225997297, 13.27991901401338177920979096245

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