Properties

Label 4-2550e2-1.1-c1e2-0-10
Degree $4$
Conductor $6502500$
Sign $1$
Analytic cond. $414.605$
Root an. cond. $4.51241$
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  − 4-s − 9-s + 8·11-s + 16-s − 8·19-s − 12·29-s − 16·31-s + 36-s + 16·41-s − 8·44-s + 10·49-s − 12·59-s + 4·61-s − 64-s − 20·71-s + 8·76-s − 8·79-s + 81-s − 12·89-s − 8·99-s − 24·101-s + 4·109-s + 12·116-s + 26·121-s + 16·124-s + 127-s + 131-s + ⋯
L(s)  = 1  − 1/2·4-s − 1/3·9-s + 2.41·11-s + 1/4·16-s − 1.83·19-s − 2.22·29-s − 2.87·31-s + 1/6·36-s + 2.49·41-s − 1.20·44-s + 10/7·49-s − 1.56·59-s + 0.512·61-s − 1/8·64-s − 2.37·71-s + 0.917·76-s − 0.900·79-s + 1/9·81-s − 1.27·89-s − 0.804·99-s − 2.38·101-s + 0.383·109-s + 1.11·116-s + 2.36·121-s + 1.43·124-s + 0.0887·127-s + 0.0873·131-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6502500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6502500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(6502500\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{4} \cdot 17^{2}\)
Sign: $1$
Analytic conductor: \(414.605\)
Root analytic conductor: \(4.51241\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 6502500,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.056144706\)
\(L(\frac12)\) \(\approx\) \(1.056144706\)
\(L(\frac{3}{2})\) 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 \( 1 \)
17$C_2$ \( 1 + T^{2} \)
good7$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - p T^{2} )^{2} \)
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$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 - 38 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 82 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
59$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 + 90 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \)
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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

−9.164211803023117544138221586685, −8.803550117426026840946027100020, −8.669883095721105648976007691226, −7.77166329164970050362670181309, −7.66106952096833165147610140496, −7.05649992689755557857076166706, −6.97847923159231804340582986809, −6.22569187672602091647602151710, −6.00836077780445760355601057727, −5.71744137030866999183655710444, −5.29365748304624770580429840224, −4.57577009366423605685451014747, −4.03894930364135756807437421118, −3.96180310079848574260274650114, −3.77743546023904913097784416572, −2.93054097669083366032716287060, −2.35826950617406455265896024736, −1.56828725657156285739851042991, −1.52976664290050473153125459204, −0.34236392066307127855807273105, 0.34236392066307127855807273105, 1.52976664290050473153125459204, 1.56828725657156285739851042991, 2.35826950617406455265896024736, 2.93054097669083366032716287060, 3.77743546023904913097784416572, 3.96180310079848574260274650114, 4.03894930364135756807437421118, 4.57577009366423605685451014747, 5.29365748304624770580429840224, 5.71744137030866999183655710444, 6.00836077780445760355601057727, 6.22569187672602091647602151710, 6.97847923159231804340582986809, 7.05649992689755557857076166706, 7.66106952096833165147610140496, 7.77166329164970050362670181309, 8.669883095721105648976007691226, 8.803550117426026840946027100020, 9.164211803023117544138221586685

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