Properties

Degree $4$
Conductor $2952$
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 − 2·3-s + 2·4-s − 2·5-s + 4·6-s + 9-s + 4·10-s − 3·11-s − 4·12-s − 6·13-s + 4·15-s − 4·16-s − 7·17-s − 2·18-s + 2·19-s − 4·20-s + 6·22-s + 6·23-s − 4·25-s + 12·26-s + 4·27-s − 3·29-s − 8·30-s + 31-s + 8·32-s + 6·33-s + 14·34-s + ⋯
L(s)  = 1  − 1.41·2-s − 1.15·3-s + 4-s − 0.894·5-s + 1.63·6-s + 1/3·9-s + 1.26·10-s − 0.904·11-s − 1.15·12-s − 1.66·13-s + 1.03·15-s − 16-s − 1.69·17-s − 0.471·18-s + 0.458·19-s − 0.894·20-s + 1.27·22-s + 1.25·23-s − 4/5·25-s + 2.35·26-s + 0.769·27-s − 0.557·29-s − 1.46·30-s + 0.179·31-s + 1.41·32-s + 1.04·33-s + 2.40·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2952 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2952 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2952\)    =    \(2^{3} \cdot 3^{2} \cdot 41\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{2952} (1, \cdot )$
Sato-Tate group: $\mathrm{USp}(4)$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 2952,\ (\ :1/2, 1/2),\ -1)\)

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)$
bad2$C_2$ \( 1 + p T + p T^{2} \)
3$C_2$ \( 1 + 2 T + p T^{2} \)
41$C_1$$\times$$C_2$ \( ( 1 + T )( 1 - 8 T + p T^{2} ) \)
good5$D_{4}$ \( 1 + 2 T + 8 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
7$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
11$C_2^2$ \( 1 + 3 T - 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 + 6 T + 30 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
17$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
19$D_{4}$ \( 1 - 2 T + 10 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 - 6 T + 20 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 + 3 T + 28 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 - T - 42 T^{2} - p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 - T - 48 T^{2} - p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 - T + 58 T^{2} - p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 - T - 54 T^{2} - p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 26 T^{2} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 6 T - 2 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 7 T + 72 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
67$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
71$D_{4}$ \( 1 - 5 T + 74 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 9 T + 116 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
79$C_2$$\times$$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
83$D_{4}$ \( 1 + 20 T + 248 T^{2} + 20 p T^{3} + p^{2} T^{4} \)
89$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + p T^{2} ) \)
97$D_{4}$ \( 1 - 4 T - 62 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
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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

−18.3503277338, −17.9148163167, −17.5259033495, −17.0012157799, −16.7643977959, −15.9637617656, −15.6450992796, −15.2111195689, −14.4199390782, −13.5358609646, −12.9488926569, −12.2781572850, −11.6743659038, −11.1919257191, −10.8036152630, −10.1876175170, −9.44683327871, −8.91591034622, −8.06088596341, −7.39062112041, −7.10629806976, −6.05005024883, −4.99826769921, −4.44567663951, −2.56662903492, 0, 2.56662903492, 4.44567663951, 4.99826769921, 6.05005024883, 7.10629806976, 7.39062112041, 8.06088596341, 8.91591034622, 9.44683327871, 10.1876175170, 10.8036152630, 11.1919257191, 11.6743659038, 12.2781572850, 12.9488926569, 13.5358609646, 14.4199390782, 15.2111195689, 15.6450992796, 15.9637617656, 16.7643977959, 17.0012157799, 17.5259033495, 17.9148163167, 18.3503277338

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