Properties

Label 4-6720e2-1.1-c1e2-0-3
Degree $4$
Conductor $45158400$
Sign $1$
Analytic cond. $2879.33$
Root an. cond. $7.32526$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·7-s − 9-s − 12·17-s + 16·23-s − 25-s − 8·31-s + 20·41-s − 16·47-s + 3·49-s + 2·63-s + 20·73-s − 32·79-s + 81-s + 12·89-s + 36·97-s + 8·103-s − 4·113-s + 24·119-s + 18·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 12·153-s + 157-s + ⋯
L(s)  = 1  − 0.755·7-s − 1/3·9-s − 2.91·17-s + 3.33·23-s − 1/5·25-s − 1.43·31-s + 3.12·41-s − 2.33·47-s + 3/7·49-s + 0.251·63-s + 2.34·73-s − 3.60·79-s + 1/9·81-s + 1.27·89-s + 3.65·97-s + 0.788·103-s − 0.376·113-s + 2.20·119-s + 1.63·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.970·153-s + 0.0798·157-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 45158400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45158400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(45158400\)    =    \(2^{12} \cdot 3^{2} \cdot 5^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(2879.33\)
Root analytic conductor: \(7.32526\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 45158400,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.212675698\)
\(L(\frac12)\) \(\approx\) \(2.212675698\)
\(L(\frac{3}{2})\) 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 \( 1 \)
3$C_2$ \( 1 + T^{2} \)
5$C_2$ \( 1 + T^{2} \)
7$C_1$ \( ( 1 + T )^{2} \)
good11$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \)
13$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
17$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - p T^{2} )^{2} \)
23$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
31$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \)
47$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
53$C_2^2$ \( 1 + 38 T^{2} + p^{2} T^{4} \)
59$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 118 T^{2} + p^{2} T^{4} \)
67$C_2^2$ \( 1 - 98 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 + 16 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 + 90 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 18 T + p T^{2} )^{2} \)
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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

−8.334388265573136937489058497762, −7.63259678908271235080984433379, −7.28141944209226053216525728084, −7.20040400440062070686925944980, −6.74032325401798585123787512812, −6.47129833503803145755219232093, −6.05722248745136339412211348290, −5.87111822365527460047713059444, −5.15125163811281334739051230500, −5.01176123547049195518397629382, −4.49764116541802413441047088972, −4.35592556297147507791697561793, −3.75183465723690479950305040347, −3.27742383104368442431705190887, −2.96020230427152335114375846518, −2.62753095166095048566623436722, −2.00289962955778904517413184899, −1.78479234855519610628042286265, −0.72709939790991479305033830099, −0.51985775397015583288991586269, 0.51985775397015583288991586269, 0.72709939790991479305033830099, 1.78479234855519610628042286265, 2.00289962955778904517413184899, 2.62753095166095048566623436722, 2.96020230427152335114375846518, 3.27742383104368442431705190887, 3.75183465723690479950305040347, 4.35592556297147507791697561793, 4.49764116541802413441047088972, 5.01176123547049195518397629382, 5.15125163811281334739051230500, 5.87111822365527460047713059444, 6.05722248745136339412211348290, 6.47129833503803145755219232093, 6.74032325401798585123787512812, 7.20040400440062070686925944980, 7.28141944209226053216525728084, 7.63259678908271235080984433379, 8.334388265573136937489058497762

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