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Properties

Label 4-2952e2-1.1-c0e2-0-1
Degree $4$
Conductor $8714304$
Sign $1$
Analytic cond. $2.17043$
Root an. cond. $1.21377$
Motivic weight $0$
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 + 2·11-s + 16-s − 2·17-s − 2·19-s − 2·25-s − 2·44-s − 64-s + 2·67-s + 2·68-s + 2·76-s + 4·83-s − 2·89-s − 2·97-s + 2·100-s + 4·107-s + 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + ⋯
L(s)  = 1  − 4-s + 2·11-s + 16-s − 2·17-s − 2·19-s − 2·25-s − 2·44-s − 64-s + 2·67-s + 2·68-s + 2·76-s + 4·83-s − 2·89-s − 2·97-s + 2·100-s + 4·107-s + 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8714304 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8714304 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(8714304\)    =    \(2^{6} \cdot 3^{4} \cdot 41^{2}\)
Sign: $1$
Analytic conductor: \(2.17043\)
Root analytic conductor: \(1.21377\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 8714304,\ (\ :0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7934656563\)
\(L(\frac12)\) \(\approx\) \(0.7934656563\)
\(L(1)\) 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 \( 1 \)
41$C_2$ \( 1 + T^{2} \)
good5$C_2$ \( ( 1 + T^{2} )^{2} \)
7$C_2^2$ \( 1 + T^{4} \)
11$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
13$C_2^2$ \( 1 + T^{4} \)
17$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
19$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
23$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
29$C_2^2$ \( 1 + T^{4} \)
31$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
37$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
43$C_2$ \( ( 1 + T^{2} )^{2} \)
47$C_2^2$ \( 1 + T^{4} \)
53$C_2^2$ \( 1 + T^{4} \)
59$C_2$ \( ( 1 + T^{2} )^{2} \)
61$C_2$ \( ( 1 + T^{2} )^{2} \)
67$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
71$C_2^2$ \( 1 + T^{4} \)
73$C_2$ \( ( 1 + T^{2} )^{2} \)
79$C_2^2$ \( 1 + T^{4} \)
83$C_1$ \( ( 1 - T )^{4} \)
89$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
97$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
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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

−9.243295826994116117665335740943, −8.782420990722053381193728192868, −8.305250158257060039593265595071, −8.277465469312798645416619415766, −7.76611344932416260310766357827, −7.05582457479912213029382548401, −6.82646301719479684737457280865, −6.49150995284329038320824611680, −6.00945627302881692355390632337, −5.91403059892012497011404907825, −5.19867276979981793555871718406, −4.65208608076711410953112643038, −4.25044645023495168417672504845, −4.23628680027043126932648062235, −3.62452224428205481999754264117, −3.42063474179712934388741844467, −2.27279303214460182383799431082, −2.13786641022128759804987783231, −1.54418040948119130240603962404, −0.56895234062587413751316224937, 0.56895234062587413751316224937, 1.54418040948119130240603962404, 2.13786641022128759804987783231, 2.27279303214460182383799431082, 3.42063474179712934388741844467, 3.62452224428205481999754264117, 4.23628680027043126932648062235, 4.25044645023495168417672504845, 4.65208608076711410953112643038, 5.19867276979981793555871718406, 5.91403059892012497011404907825, 6.00945627302881692355390632337, 6.49150995284329038320824611680, 6.82646301719479684737457280865, 7.05582457479912213029382548401, 7.76611344932416260310766357827, 8.277465469312798645416619415766, 8.305250158257060039593265595071, 8.782420990722053381193728192868, 9.243295826994116117665335740943

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