Properties

Degree $4$
Conductor $5292$
Sign $1$
Motivic weight $1$
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 4-s − 2·7-s + 9-s − 12-s + 12·13-s + 16-s − 8·19-s + 2·21-s − 6·25-s − 27-s − 2·28-s + 36-s − 20·37-s − 12·39-s − 8·43-s − 48-s + 3·49-s + 12·52-s + 8·57-s + 12·61-s − 2·63-s + 64-s + 8·67-s + 20·73-s + 6·75-s − 8·76-s + ⋯
L(s)  = 1  − 0.577·3-s + 1/2·4-s − 0.755·7-s + 1/3·9-s − 0.288·12-s + 3.32·13-s + 1/4·16-s − 1.83·19-s + 0.436·21-s − 6/5·25-s − 0.192·27-s − 0.377·28-s + 1/6·36-s − 3.28·37-s − 1.92·39-s − 1.21·43-s − 0.144·48-s + 3/7·49-s + 1.66·52-s + 1.05·57-s + 1.53·61-s − 0.251·63-s + 1/8·64-s + 0.977·67-s + 2.34·73-s + 0.692·75-s − 0.917·76-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5292 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5292 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(5292\)    =    \(2^{2} \cdot 3^{3} \cdot 7^{2}\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{5292} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 5292,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7910759084\)
\(L(\frac12)\) \(\approx\) \(0.7910759084\)
\(L(\frac{3}{2})\) 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_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
3$C_1$ \( 1 + T \)
7$C_1$ \( ( 1 + T )^{2} \)
good5$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
11$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
13$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
19$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
29$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
31$C_2$ \( ( 1 + p T^{2} )^{2} \)
37$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
43$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
61$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
73$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
97$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \)
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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

−12.25720048996309464371398882337, −11.37318927677395242391128273111, −11.12015185557948517225700432181, −10.51951977961491945205840276286, −10.13805865622749455803240285872, −9.164662289745378790951914691450, −8.356918966632091922574869118017, −8.337652051623380123011624806396, −6.84552480602986516155667793601, −6.58813712157544804919343144385, −6.01899647743800981343957937964, −5.33985014787602837985094112170, −3.83863171110134241641513411613, −3.62482887081886485478101246558, −1.79365373540820646322667624458, 1.79365373540820646322667624458, 3.62482887081886485478101246558, 3.83863171110134241641513411613, 5.33985014787602837985094112170, 6.01899647743800981343957937964, 6.58813712157544804919343144385, 6.84552480602986516155667793601, 8.337652051623380123011624806396, 8.356918966632091922574869118017, 9.164662289745378790951914691450, 10.13805865622749455803240285872, 10.51951977961491945205840276286, 11.12015185557948517225700432181, 11.37318927677395242391128273111, 12.25720048996309464371398882337

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