Properties

Label 4-425e2-1.1-c1e2-0-6
Degree $4$
Conductor $180625$
Sign $1$
Analytic cond. $11.5168$
Root an. cond. $1.84218$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·4-s + 6·9-s + 5·16-s + 8·19-s − 12·29-s + 8·31-s + 18·36-s − 12·41-s − 2·49-s + 24·59-s − 20·61-s + 3·64-s − 8·71-s + 24·76-s − 24·79-s + 27·81-s − 20·89-s − 20·101-s − 12·109-s − 36·116-s − 22·121-s + 24·124-s + 127-s + 131-s + 137-s + 139-s + 30·144-s + ⋯
L(s)  = 1  + 3/2·4-s + 2·9-s + 5/4·16-s + 1.83·19-s − 2.22·29-s + 1.43·31-s + 3·36-s − 1.87·41-s − 2/7·49-s + 3.12·59-s − 2.56·61-s + 3/8·64-s − 0.949·71-s + 2.75·76-s − 2.70·79-s + 3·81-s − 2.11·89-s − 1.99·101-s − 1.14·109-s − 3.34·116-s − 2·121-s + 2.15·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 5/2·144-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 180625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 180625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(180625\)    =    \(5^{4} \cdot 17^{2}\)
Sign: $1$
Analytic conductor: \(11.5168\)
Root analytic conductor: \(1.84218\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 180625,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.015633321\)
\(L(\frac12)\) \(\approx\) \(3.015633321\)
\(L(\frac{3}{2})\) 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)$
bad5 \( 1 \)
17$C_2$ \( 1 + T^{2} \)
good2$C_2^2$ \( 1 - 3 T^{2} + p^{2} T^{4} \)
3$C_2$ \( ( 1 - p T^{2} )^{2} \)
7$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
41$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
47$C_2$ \( ( 1 - p T^{2} )^{2} \)
53$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 118 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
79$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 150 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 - 190 T^{2} + p^{2} T^{4} \)
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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

−11.30343291001595261223671925186, −11.12545955184019786026691703065, −10.34419994153612588694837079470, −10.20379975024811324405866717620, −9.580103581843853268273104393610, −9.526145800625271601965973157201, −8.571556531784596602191283720981, −8.054755129932527360045391104644, −7.37323907538424822017392644714, −7.30700044211496192342148109860, −6.88283093269140417083303385877, −6.43689548753242383069829388258, −5.64558969832405997965298070006, −5.34179206358898239150626680022, −4.50562702493167937967098019859, −3.96408590383967063163987482962, −3.27421846633199865520241686865, −2.67920169036367087688039689822, −1.66415178220154260649314063962, −1.39699613843806692195408059085, 1.39699613843806692195408059085, 1.66415178220154260649314063962, 2.67920169036367087688039689822, 3.27421846633199865520241686865, 3.96408590383967063163987482962, 4.50562702493167937967098019859, 5.34179206358898239150626680022, 5.64558969832405997965298070006, 6.43689548753242383069829388258, 6.88283093269140417083303385877, 7.30700044211496192342148109860, 7.37323907538424822017392644714, 8.054755129932527360045391104644, 8.571556531784596602191283720981, 9.526145800625271601965973157201, 9.580103581843853268273104393610, 10.20379975024811324405866717620, 10.34419994153612588694837079470, 11.12545955184019786026691703065, 11.30343291001595261223671925186

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