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Properties

Label 4-538e2-1.1-c1e2-0-1
Degree $4$
Conductor $289444$
Sign $1$
Analytic cond. $18.4552$
Root an. cond. $2.07266$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s − 3-s + 3·4-s − 5·5-s − 2·6-s − 4·7-s + 4·8-s − 4·9-s − 10·10-s − 4·11-s − 3·12-s − 2·13-s − 8·14-s + 5·15-s + 5·16-s − 4·17-s − 8·18-s + 4·19-s − 15·20-s + 4·21-s − 8·22-s − 12·23-s − 4·24-s + 10·25-s − 4·26-s + 6·27-s − 12·28-s + ⋯
L(s)  = 1  + 1.41·2-s − 0.577·3-s + 3/2·4-s − 2.23·5-s − 0.816·6-s − 1.51·7-s + 1.41·8-s − 4/3·9-s − 3.16·10-s − 1.20·11-s − 0.866·12-s − 0.554·13-s − 2.13·14-s + 1.29·15-s + 5/4·16-s − 0.970·17-s − 1.88·18-s + 0.917·19-s − 3.35·20-s + 0.872·21-s − 1.70·22-s − 2.50·23-s − 0.816·24-s + 2·25-s − 0.784·26-s + 1.15·27-s − 2.26·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 289444 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289444 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(289444\)    =    \(2^{2} \cdot 269^{2}\)
Sign: $1$
Analytic conductor: \(18.4552\)
Root analytic conductor: \(2.07266\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 289444,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{2} \)
269$C_1$ \( ( 1 - T )^{2} \)
good3$D_{4}$ \( 1 + T + 5 T^{2} + p T^{3} + p^{2} T^{4} \)
5$C_4$ \( 1 + p T + 3 p T^{2} + p^{2} T^{3} + p^{2} T^{4} \)
7$D_{4}$ \( 1 + 4 T + 13 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 + 4 T + 21 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
17$D_{4}$ \( 1 + 4 T + 33 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
23$D_{4}$ \( 1 + 12 T + 77 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 + 5 T + 53 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 - 6 T + 51 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 + 4 T + 33 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 - 2 T + 38 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
43$C_2^2$ \( 1 + 81 T^{2} + p^{2} T^{4} \)
47$D_{4}$ \( 1 + 7 T + 95 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
53$C_2$ \( ( 1 + p T^{2} )^{2} \)
59$D_{4}$ \( 1 - 7 T + 29 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 2 T + 103 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 - 8 T + 145 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 - 6 T + 106 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 8 T + 117 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 3 T + 9 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + 6 T + 170 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 - 5 T + 33 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 6 T - 42 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
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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

−10.75550022492265920355365148009, −10.48423796127544971822527316674, −9.779310370633365036992065654641, −9.522969433472425107289092431849, −8.409896087057840088796039549985, −8.340542059575305887345664585683, −7.65757781263824684391871602608, −7.55313253354201825755994795926, −6.69189875945640086358163731787, −6.52286239153267665937487243535, −5.78771380616476145548109461474, −5.53522707827406947590541907383, −4.88465287099634814764121847312, −4.34816336712005722797211848473, −3.72426930725569106217054384489, −3.49314352731394953195257893184, −2.81533988877349973915721995346, −2.34630354444356126931490073198, 0, 0, 2.34630354444356126931490073198, 2.81533988877349973915721995346, 3.49314352731394953195257893184, 3.72426930725569106217054384489, 4.34816336712005722797211848473, 4.88465287099634814764121847312, 5.53522707827406947590541907383, 5.78771380616476145548109461474, 6.52286239153267665937487243535, 6.69189875945640086358163731787, 7.55313253354201825755994795926, 7.65757781263824684391871602608, 8.340542059575305887345664585683, 8.409896087057840088796039549985, 9.522969433472425107289092431849, 9.779310370633365036992065654641, 10.48423796127544971822527316674, 10.75550022492265920355365148009

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