Properties

Label 4-25047-1.1-c1e2-0-1
Degree $4$
Conductor $25047$
Sign $-1$
Analytic cond. $1.59701$
Root an. cond. $1.12415$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 4-s − 5·5-s − 2·9-s − 5·11-s + 12-s − 5·15-s − 3·16-s − 5·20-s + 7·23-s + 9·25-s − 5·27-s − 5·31-s − 5·33-s − 2·36-s + 37-s − 5·44-s + 10·45-s − 15·47-s − 3·48-s + 9·49-s − 5·53-s + 25·55-s + 5·59-s − 5·60-s − 7·64-s − 6·67-s + ⋯
L(s)  = 1  + 0.577·3-s + 1/2·4-s − 2.23·5-s − 2/3·9-s − 1.50·11-s + 0.288·12-s − 1.29·15-s − 3/4·16-s − 1.11·20-s + 1.45·23-s + 9/5·25-s − 0.962·27-s − 0.898·31-s − 0.870·33-s − 1/3·36-s + 0.164·37-s − 0.753·44-s + 1.49·45-s − 2.18·47-s − 0.433·48-s + 9/7·49-s − 0.686·53-s + 3.37·55-s + 0.650·59-s − 0.645·60-s − 7/8·64-s − 0.733·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 25047 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25047 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(25047\)    =    \(3^{2} \cdot 11^{2} \cdot 23\)
Sign: $-1$
Analytic conductor: \(1.59701\)
Root analytic conductor: \(1.12415\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 25047,\ (\ :1/2, 1/2),\ -1)\)

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$C_2$ \( 1 - T + p T^{2} \)
11$C_2$ \( 1 + 5 T + p T^{2} \)
23$C_1$$\times$$C_2$ \( ( 1 - T )( 1 - 6 T + p T^{2} ) \)
good2$C_2^2$ \( 1 - T^{2} + p^{2} T^{4} \)
5$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
7$C_2^2$ \( 1 - 9 T^{2} + p^{2} T^{4} \)
13$C_2^2$ \( 1 - T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 + 24 T^{2} + p^{2} T^{4} \)
19$C_2^2$ \( 1 - 3 T^{2} + p^{2} T^{4} \)
29$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
31$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 5 T + p T^{2} ) \)
37$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
41$C_2^2$ \( 1 - 43 T^{2} + p^{2} T^{4} \)
43$C_2^2$ \( 1 + 29 T^{2} + p^{2} T^{4} \)
47$C_2$$\times$$C_2$ \( ( 1 + 3 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
53$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 9 T + p T^{2} ) \)
59$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
61$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
67$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
71$C_2$$\times$$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
73$C_2^2$ \( 1 - 101 T^{2} + p^{2} T^{4} \)
79$C_2^2$ \( 1 - 123 T^{2} + p^{2} T^{4} \)
83$C_2^2$ \( 1 - 49 T^{2} + p^{2} T^{4} \)
89$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 - 6 T + p T^{2} ) \)
97$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
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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

−10.74179251154727516089599523799, −9.902755497773684101105287373651, −9.106774912378504913663011839273, −8.687542931500245289982807739994, −8.100402166539905161326145110748, −7.73064417483830706166950601404, −7.36245284015338742118544299322, −6.80060567458629117471687557978, −5.86688197550175256119713199520, −5.04737094667169973548022377791, −4.50553616067697250442961945989, −3.56837153068237509265006210769, −3.15063744642578337967576184396, −2.31868685095382966799799445864, 0, 2.31868685095382966799799445864, 3.15063744642578337967576184396, 3.56837153068237509265006210769, 4.50553616067697250442961945989, 5.04737094667169973548022377791, 5.86688197550175256119713199520, 6.80060567458629117471687557978, 7.36245284015338742118544299322, 7.73064417483830706166950601404, 8.100402166539905161326145110748, 8.687542931500245289982807739994, 9.106774912378504913663011839273, 9.902755497773684101105287373651, 10.74179251154727516089599523799

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