Properties

Degree $4$
Conductor $2457$
Sign $-1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s − 3-s + 4-s − 2·5-s + 2·6-s − 3·7-s + 9-s + 4·10-s − 12-s − 7·13-s + 6·14-s + 2·15-s + 16-s − 2·18-s − 2·20-s + 3·21-s − 2·23-s + 2·25-s + 14·26-s − 27-s − 3·28-s + 2·29-s − 4·30-s + 4·31-s + 2·32-s + 6·35-s + 36-s + ⋯
L(s)  = 1  − 1.41·2-s − 0.577·3-s + 1/2·4-s − 0.894·5-s + 0.816·6-s − 1.13·7-s + 1/3·9-s + 1.26·10-s − 0.288·12-s − 1.94·13-s + 1.60·14-s + 0.516·15-s + 1/4·16-s − 0.471·18-s − 0.447·20-s + 0.654·21-s − 0.417·23-s + 2/5·25-s + 2.74·26-s − 0.192·27-s − 0.566·28-s + 0.371·29-s − 0.730·30-s + 0.718·31-s + 0.353·32-s + 1.01·35-s + 1/6·36-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2457 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2457 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2457\)    =    \(3^{3} \cdot 7 \cdot 13\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{2457} (1, \cdot )$
Sato-Tate group: $\mathrm{USp}(4)$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 2457,\ (\ :1/2, 1/2),\ -1)\)

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$C_1$ \( 1 + T \)
7$C_1$$\times$$C_2$ \( ( 1 - T )( 1 + 4 T + p T^{2} ) \)
13$C_1$$\times$$C_2$ \( ( 1 + T )( 1 + 6 T + p T^{2} ) \)
good2$D_{4}$ \( 1 + p T + 3 T^{2} + p^{2} T^{3} + p^{2} T^{4} \)
5$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
11$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
23$D_{4}$ \( 1 + 2 T + 6 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
29$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
31$D_{4}$ \( 1 - 4 T - 2 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
37$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
41$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
43$C_2^2$ \( 1 + 38 T^{2} + p^{2} T^{4} \)
47$D_{4}$ \( 1 - 2 T + 38 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 - 2 T - 46 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
59$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
61$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
67$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
71$D_{4}$ \( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 12 T + 70 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
83$D_{4}$ \( 1 - 2 T + 22 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 - 10 T + 162 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
97$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 18 T + p T^{2} ) \)
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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

−18.8497507948, −18.3394428734, −17.6842669463, −17.2789389949, −16.7871167131, −16.4641016883, −15.6553589693, −15.3830132091, −14.7353317118, −13.9169376774, −13.2019051783, −12.4303995839, −12.0451117480, −11.6973573963, −10.5645715515, −10.2017468926, −9.59094782975, −9.19529109569, −8.29858964030, −7.73184051343, −7.00738197527, −6.36719361677, −5.20217380021, −4.26142870616, −2.93181472338, 0, 2.93181472338, 4.26142870616, 5.20217380021, 6.36719361677, 7.00738197527, 7.73184051343, 8.29858964030, 9.19529109569, 9.59094782975, 10.2017468926, 10.5645715515, 11.6973573963, 12.0451117480, 12.4303995839, 13.2019051783, 13.9169376774, 14.7353317118, 15.3830132091, 15.6553589693, 16.4641016883, 16.7871167131, 17.2789389949, 17.6842669463, 18.3394428734, 18.8497507948

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