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Properties

Label 4-206e2-1.1-c1e2-0-2
Degree $4$
Conductor $42436$
Sign $1$
Analytic cond. $2.70575$
Root an. cond. $1.28254$
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-s + 4·3-s + 2·5-s + 4·6-s − 8-s + 6·9-s + 2·10-s − 4·13-s + 8·15-s − 16-s − 5·17-s + 6·18-s − 2·19-s − 6·23-s − 4·24-s + 5·25-s − 4·26-s − 4·27-s + 8·30-s + 10·31-s − 5·34-s + 4·37-s − 2·38-s − 16·39-s − 2·40-s − 5·41-s − 2·43-s + ⋯
L(s)  = 1  + 0.707·2-s + 2.30·3-s + 0.894·5-s + 1.63·6-s − 0.353·8-s + 2·9-s + 0.632·10-s − 1.10·13-s + 2.06·15-s − 1/4·16-s − 1.21·17-s + 1.41·18-s − 0.458·19-s − 1.25·23-s − 0.816·24-s + 25-s − 0.784·26-s − 0.769·27-s + 1.46·30-s + 1.79·31-s − 0.857·34-s + 0.657·37-s − 0.324·38-s − 2.56·39-s − 0.316·40-s − 0.780·41-s − 0.304·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 42436 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 42436 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(42436\)    =    \(2^{2} \cdot 103^{2}\)
Sign: $1$
Analytic conductor: \(2.70575\)
Root analytic conductor: \(1.28254\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 42436,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.683283228\)
\(L(\frac12)\) \(\approx\) \(3.683283228\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - T + T^{2} \)
103$C_2$ \( 1 - 20 T + p T^{2} \)
good3$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
5$C_2^2$ \( 1 - 2 T - T^{2} - 2 p T^{3} + p^{2} T^{4} \)
7$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
17$C_2^2$ \( 1 + 5 T + 8 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
19$C_2^2$ \( 1 + 2 T - 15 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
23$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
29$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
31$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 + 5 T - 16 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
43$C_2^2$ \( 1 + 2 T - 39 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 5 T - 22 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 12 T + 91 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 + 6 T - 23 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
61$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 + 10 T + 33 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
71$C_2^2$ \( 1 - 3 T - 62 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
73$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 + 14 T + 113 T^{2} + 14 p T^{3} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 11 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 - 5 T + p T^{2} ) \)
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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

−12.94183478188430511938989955808, −12.36278083495067501489694967266, −11.83703969510683728897979230799, −11.35932774178449581880084840330, −10.26492441486943179318037494496, −10.25053732357731402487780095875, −9.525372906212732530663740840033, −9.116438464983217811635637083347, −8.569330729453508916382359872209, −8.500569149411759326659108716548, −7.61145799850211050764669888869, −7.29902487470949888835688159897, −6.28544167734757414228536079061, −6.04674917699901412033587612725, −5.03506232483960009380372706301, −4.47967091578696093696330695786, −3.81353668984295986008598968921, −3.05277376291281144082194601118, −2.28114304826638684983775902776, −2.25986991265905305405720668491, 2.25986991265905305405720668491, 2.28114304826638684983775902776, 3.05277376291281144082194601118, 3.81353668984295986008598968921, 4.47967091578696093696330695786, 5.03506232483960009380372706301, 6.04674917699901412033587612725, 6.28544167734757414228536079061, 7.29902487470949888835688159897, 7.61145799850211050764669888869, 8.500569149411759326659108716548, 8.569330729453508916382359872209, 9.116438464983217811635637083347, 9.525372906212732530663740840033, 10.25053732357731402487780095875, 10.26492441486943179318037494496, 11.35932774178449581880084840330, 11.83703969510683728897979230799, 12.36278083495067501489694967266, 12.94183478188430511938989955808

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