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Properties

Label 4-3328e2-1.1-c0e2-0-1
Degree $4$
Conductor $11075584$
Sign $1$
Analytic cond. $2.75855$
Root an. cond. $1.28875$
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  − 2·5-s − 2·9-s − 2·13-s + 2·25-s + 4·29-s − 2·37-s + 2·41-s + 4·45-s + 4·65-s + 2·73-s + 3·81-s − 2·89-s − 2·97-s − 2·109-s + 4·117-s − 2·125-s + 127-s + 131-s + 137-s + 139-s − 8·145-s + 149-s + 151-s + 157-s + 163-s + 167-s + 3·169-s + ⋯
L(s)  = 1  − 2·5-s − 2·9-s − 2·13-s + 2·25-s + 4·29-s − 2·37-s + 2·41-s + 4·45-s + 4·65-s + 2·73-s + 3·81-s − 2·89-s − 2·97-s − 2·109-s + 4·117-s − 2·125-s + 127-s + 131-s + 137-s + 139-s − 8·145-s + 149-s + 151-s + 157-s + 163-s + 167-s + 3·169-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 11075584 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11075584 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(11075584\)    =    \(2^{16} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(2.75855\)
Root analytic conductor: \(1.28875\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 11075584,\ (\ :0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3309935908\)
\(L(\frac12)\) \(\approx\) \(0.3309935908\)
\(L(1)\) 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 \( 1 \)
13$C_1$ \( ( 1 + T )^{2} \)
good3$C_2$ \( ( 1 + T^{2} )^{2} \)
5$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
7$C_2^2$ \( 1 + T^{4} \)
11$C_2^2$ \( 1 + T^{4} \)
17$C_2$ \( ( 1 + T^{2} )^{2} \)
19$C_2^2$ \( 1 + T^{4} \)
23$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
29$C_1$ \( ( 1 - T )^{4} \)
31$C_2^2$ \( 1 + T^{4} \)
37$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
41$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
43$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
47$C_2^2$ \( 1 + T^{4} \)
53$C_2$ \( ( 1 + T^{2} )^{2} \)
59$C_2^2$ \( 1 + T^{4} \)
61$C_2$ \( ( 1 + T^{2} )^{2} \)
67$C_2^2$ \( 1 + T^{4} \)
71$C_2^2$ \( 1 + T^{4} \)
73$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
79$C_2$ \( ( 1 + T^{2} )^{2} \)
83$C_2^2$ \( 1 + T^{4} \)
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

−8.683566104238731047272374834983, −8.582985594314454595550991724989, −8.242141702300957214717329453649, −7.81520521953101285156699409397, −7.80401936377682491840190525021, −7.12030323217604796249396735785, −6.72916288767080884441788063230, −6.66650127388659503924260990196, −5.94567037741323216314380736925, −5.53177382797696908474011046011, −4.97240062320504242813866530914, −4.86653243440099295009878688818, −4.43827007065705420078543802342, −3.86651587894082261924268524192, −3.53349567947064005915898373053, −2.88168046444037623653313610715, −2.56133958947634837638968183931, −2.53719421717224670509687213378, −1.18954866088576765547196896422, −0.37409227542500695295941811801, 0.37409227542500695295941811801, 1.18954866088576765547196896422, 2.53719421717224670509687213378, 2.56133958947634837638968183931, 2.88168046444037623653313610715, 3.53349567947064005915898373053, 3.86651587894082261924268524192, 4.43827007065705420078543802342, 4.86653243440099295009878688818, 4.97240062320504242813866530914, 5.53177382797696908474011046011, 5.94567037741323216314380736925, 6.66650127388659503924260990196, 6.72916288767080884441788063230, 7.12030323217604796249396735785, 7.80401936377682491840190525021, 7.81520521953101285156699409397, 8.242141702300957214717329453649, 8.582985594314454595550991724989, 8.683566104238731047272374834983

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