Properties

Label 4-15e4-1.1-c1e2-0-9
Degree $4$
Conductor $50625$
Sign $1$
Analytic cond. $3.22789$
Root an. cond. $1.34038$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Related objects

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·4-s − 10·7-s − 10·13-s + 12·16-s − 2·19-s + 40·28-s − 14·31-s + 20·37-s − 10·43-s + 61·49-s + 40·52-s − 26·61-s − 32·64-s − 10·67-s + 20·73-s + 8·76-s − 8·79-s + 100·91-s − 10·97-s − 40·103-s − 38·109-s − 120·112-s − 22·121-s + 56·124-s + 127-s + 131-s + 20·133-s + ⋯
L(s)  = 1  − 2·4-s − 3.77·7-s − 2.77·13-s + 3·16-s − 0.458·19-s + 7.55·28-s − 2.51·31-s + 3.28·37-s − 1.52·43-s + 61/7·49-s + 5.54·52-s − 3.32·61-s − 4·64-s − 1.22·67-s + 2.34·73-s + 0.917·76-s − 0.900·79-s + 10.4·91-s − 1.01·97-s − 3.94·103-s − 3.63·109-s − 11.3·112-s − 2·121-s + 5.02·124-s + 0.0887·127-s + 0.0873·131-s + 1.73·133-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 50625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(50625\)    =    \(3^{4} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(3.22789\)
Root analytic conductor: \(1.34038\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 50625,\ (\ :1/2, 1/2),\ 1)\)

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 \( 1 \)
5 \( 1 \)
good2$C_2$ \( ( 1 + p T^{2} )^{2} \)
7$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
11$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 + p T^{2} )^{2} \)
19$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2$ \( ( 1 + p T^{2} )^{2} \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 13 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 + p T^{2} )^{2} \)
89$C_2$ \( ( 1 + p T^{2} )^{2} \)
97$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
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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

−15.30147419110247024970488501454, −14.48091991161374155122203789126, −14.48091991161374155122203789126, −13.39726139601378948319375315570, −13.39726139601378948319375315570, −12.81633965924763019228170272669, −12.81633965924763019228170272669, −12.11951399488920175322797663822, −12.11951399488920175322797663822, −10.49393585601286522315094653400, −10.49393585601286522315094653400, −9.599576350131101649309623215930, −9.599576350131101649309623215930, −9.206590206514279856778701903578, −9.206590206514279856778701903578, −7.75954149302713039814888795826, −7.75954149302713039814888795826, −6.62145632893118011347307968766, −6.62145632893118011347307968766, −5.46981299804941090347456591942, −5.46981299804941090347456591942, −4.14585724745469274814887485348, −4.14585724745469274814887485348, −2.93248946372060750659719660133, −2.93248946372060750659719660133, 0, 0, 2.93248946372060750659719660133, 2.93248946372060750659719660133, 4.14585724745469274814887485348, 4.14585724745469274814887485348, 5.46981299804941090347456591942, 5.46981299804941090347456591942, 6.62145632893118011347307968766, 6.62145632893118011347307968766, 7.75954149302713039814888795826, 7.75954149302713039814888795826, 9.206590206514279856778701903578, 9.206590206514279856778701903578, 9.599576350131101649309623215930, 9.599576350131101649309623215930, 10.49393585601286522315094653400, 10.49393585601286522315094653400, 12.11951399488920175322797663822, 12.11951399488920175322797663822, 12.81633965924763019228170272669, 12.81633965924763019228170272669, 13.39726139601378948319375315570, 13.39726139601378948319375315570, 14.48091991161374155122203789126, 14.48091991161374155122203789126, 15.30147419110247024970488501454

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