Properties

Label 4-1134e2-1.1-c1e2-0-38
Degree $4$
Conductor $1285956$
Sign $1$
Analytic cond. $81.9936$
Root an. cond. $3.00915$
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  + 2-s + 5-s + 7-s − 8-s + 10-s + 5·11-s + 14-s − 16-s − 4·17-s − 2·19-s + 5·22-s − 23-s + 5·25-s + 4·29-s + 9·31-s − 4·34-s + 35-s + 10·37-s − 2·38-s − 40-s − 9·41-s + 10·43-s − 46-s + 6·47-s + 5·50-s − 24·53-s + 5·55-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.447·5-s + 0.377·7-s − 0.353·8-s + 0.316·10-s + 1.50·11-s + 0.267·14-s − 1/4·16-s − 0.970·17-s − 0.458·19-s + 1.06·22-s − 0.208·23-s + 25-s + 0.742·29-s + 1.61·31-s − 0.685·34-s + 0.169·35-s + 1.64·37-s − 0.324·38-s − 0.158·40-s − 1.40·41-s + 1.52·43-s − 0.147·46-s + 0.875·47-s + 0.707·50-s − 3.29·53-s + 0.674·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1285956 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1285956 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1285956\)    =    \(2^{2} \cdot 3^{8} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(81.9936\)
Root analytic conductor: \(3.00915\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1285956,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.782832442\)
\(L(\frac12)\) \(\approx\) \(3.782832442\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - T + T^{2} \)
3 \( 1 \)
7$C_2$ \( 1 - T + T^{2} \)
good5$C_2^2$ \( 1 - T - 4 T^{2} - p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - 5 T + 14 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
13$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
17$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 + T - 22 T^{2} + p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 - 4 T - 13 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
31$C_2^2$ \( 1 - 9 T + 50 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
37$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 + 9 T + 40 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
43$C_2^2$ \( 1 - 10 T + 57 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 6 T - 11 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
53$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 + 14 T + 137 T^{2} + 14 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
67$C_2^2$ \( 1 - 8 T - 3 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 13 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
79$C_2^2$ \( 1 + 6 T - 43 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
83$C_2^2$ \( 1 + 4 T - 67 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 + 16 T + 159 T^{2} + 16 p T^{3} + p^{2} T^{4} \)
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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

−9.740096032895099323638588687997, −9.598678269261426001868487751775, −9.363713051091779548762518232883, −8.801317671783169534550996703550, −8.289374363277257252818871019625, −8.182027117876338988264903121633, −7.52966299640466144006561091254, −6.78665180005330902717845013139, −6.60321627284914591966157330330, −6.27444471439716991336630602461, −5.92230037174648159866103124279, −5.20602691108412793123885231529, −4.72686872491089972321121914351, −4.35220577229515076364761828740, −4.16776633734292109426313256674, −3.32115426094825788779717376134, −2.87101801913459699156706697284, −2.24661924438002086857101643449, −1.54734269339244836099079402897, −0.812965127840567764807655029651, 0.812965127840567764807655029651, 1.54734269339244836099079402897, 2.24661924438002086857101643449, 2.87101801913459699156706697284, 3.32115426094825788779717376134, 4.16776633734292109426313256674, 4.35220577229515076364761828740, 4.72686872491089972321121914351, 5.20602691108412793123885231529, 5.92230037174648159866103124279, 6.27444471439716991336630602461, 6.60321627284914591966157330330, 6.78665180005330902717845013139, 7.52966299640466144006561091254, 8.182027117876338988264903121633, 8.289374363277257252818871019625, 8.801317671783169534550996703550, 9.363713051091779548762518232883, 9.598678269261426001868487751775, 9.740096032895099323638588687997

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