Properties

Degree $4$
Conductor $81225$
Sign $-1$
Motivic weight $1$
Primitive no
Self-dual yes
Analytic rank $1$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·5-s + 9-s + 2·11-s − 4·16-s − 2·19-s + 4·25-s − 4·29-s − 12·31-s − 3·45-s + 11·49-s − 6·55-s − 16·59-s − 2·61-s − 24·71-s + 32·79-s + 12·80-s + 81-s − 12·89-s + 6·95-s + 2·99-s + 4·101-s + 8·109-s − 19·121-s + 3·125-s + 127-s + 131-s + 137-s + ⋯
L(s)  = 1  − 1.34·5-s + 1/3·9-s + 0.603·11-s − 16-s − 0.458·19-s + 4/5·25-s − 0.742·29-s − 2.15·31-s − 0.447·45-s + 11/7·49-s − 0.809·55-s − 2.08·59-s − 0.256·61-s − 2.84·71-s + 3.60·79-s + 1.34·80-s + 1/9·81-s − 1.27·89-s + 0.615·95-s + 0.201·99-s + 0.398·101-s + 0.766·109-s − 1.72·121-s + 0.268·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 81225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(81225\)    =    \(3^{2} \cdot 5^{2} \cdot 19^{2}\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{81225} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 81225,\ (\ :1/2, 1/2),\ -1)\)

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_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
5$C_2$ \( 1 + 3 T + p T^{2} \)
19$C_1$ \( ( 1 + T )^{2} \)
good2$C_2$ \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \)
7$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
11$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
17$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
23$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
29$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 + p T^{2} )^{2} \)
41$C_2$ \( ( 1 + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
47$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
59$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
71$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 16 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
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   \(L(s) = \displaystyle\prod_{\mathfrak{p}\ \mathrm{bad}} (1- a(\mathfrak{p}) (N\mathfrak{p})^{-s})^{-1} \prod_{\mathfrak{p}\ \mathrm{good}} (1- a(\mathfrak{p}) (N\mathfrak{p})^{-s} + (N\mathfrak{p})^{-2s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.341053920747992022463934553266, −8.836427381447004052087018681826, −8.723409361965311529712840067335, −7.74191106110524052069368926798, −7.47848778675296589128891353641, −7.15886697828319609358757774448, −6.41173993136661837405396202601, −5.94366419557132992161838079855, −5.07072622007111059188732569210, −4.55783516638495121923212657411, −3.81591641511052084615077615666, −3.69330683704517818549980694450, −2.57494569661476323183620903482, −1.60475124369086001622436622243, 0, 1.60475124369086001622436622243, 2.57494569661476323183620903482, 3.69330683704517818549980694450, 3.81591641511052084615077615666, 4.55783516638495121923212657411, 5.07072622007111059188732569210, 5.94366419557132992161838079855, 6.41173993136661837405396202601, 7.15886697828319609358757774448, 7.47848778675296589128891353641, 7.74191106110524052069368926798, 8.723409361965311529712840067335, 8.836427381447004052087018681826, 9.341053920747992022463934553266

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