Properties

Label 4-1650e2-1.1-c1e2-0-15
Degree $4$
Conductor $2722500$
Sign $1$
Analytic cond. $173.588$
Root an. cond. $3.62978$
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  − 4-s − 9-s − 2·11-s + 16-s + 8·19-s − 12·29-s + 16·31-s + 36-s + 12·41-s + 2·44-s + 10·49-s + 16·61-s − 64-s + 12·71-s − 8·76-s − 28·79-s + 81-s + 12·89-s + 2·99-s + 12·101-s + 8·109-s + 12·116-s + 3·121-s − 16·124-s + 127-s + 131-s + 137-s + ⋯
L(s)  = 1  − 1/2·4-s − 1/3·9-s − 0.603·11-s + 1/4·16-s + 1.83·19-s − 2.22·29-s + 2.87·31-s + 1/6·36-s + 1.87·41-s + 0.301·44-s + 10/7·49-s + 2.04·61-s − 1/8·64-s + 1.42·71-s − 0.917·76-s − 3.15·79-s + 1/9·81-s + 1.27·89-s + 0.201·99-s + 1.19·101-s + 0.766·109-s + 1.11·116-s + 3/11·121-s − 1.43·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2722500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2722500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2722500\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{4} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(173.588\)
Root analytic conductor: \(3.62978\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1650} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 2722500,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.955676276\)
\(L(\frac12)\) \(\approx\) \(1.955676276\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T^{2} \)
3$C_2$ \( 1 + T^{2} \)
5 \( 1 \)
11$C_1$ \( ( 1 + T )^{2} \)
good7$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
17$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 - p T^{2} )^{2} \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 118 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 142 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
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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

−9.815626142798945392344320931911, −9.161493250100856088232914073754, −8.697682676434650456013254565807, −8.539656211641887564007120877719, −7.898714789356682017862785155492, −7.50472410226835804207180047752, −7.47056351506611648043844488695, −6.81727640020239548589089864255, −6.25910130489876625955844609685, −5.72604065846186135363486619021, −5.62626800481337350779750276752, −5.00495502119669651819120876736, −4.74478526447514835009412810378, −3.94382568015696916455588784502, −3.83998232499477175671902486222, −3.02920176134488095376951961213, −2.68733075069823990382751008584, −2.13343775033899613311035906501, −1.14299162909676704652205455168, −0.62995539807950837900953515777, 0.62995539807950837900953515777, 1.14299162909676704652205455168, 2.13343775033899613311035906501, 2.68733075069823990382751008584, 3.02920176134488095376951961213, 3.83998232499477175671902486222, 3.94382568015696916455588784502, 4.74478526447514835009412810378, 5.00495502119669651819120876736, 5.62626800481337350779750276752, 5.72604065846186135363486619021, 6.25910130489876625955844609685, 6.81727640020239548589089864255, 7.47056351506611648043844488695, 7.50472410226835804207180047752, 7.898714789356682017862785155492, 8.539656211641887564007120877719, 8.697682676434650456013254565807, 9.161493250100856088232914073754, 9.815626142798945392344320931911

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