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Properties

Label 4-3744e2-1.1-c0e2-0-4
Degree $4$
Conductor $14017536$
Sign $1$
Analytic cond. $3.49129$
Root an. cond. $1.36693$
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 + 2·25-s + 4·29-s + 2·37-s + 2·41-s − 4·61-s − 2·73-s − 2·89-s − 2·97-s − 2·109-s + 4·113-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 − 169-s + 173-s + 179-s + 181-s + 4·185-s + ⋯
L(s)  = 1  + 2·5-s + 2·25-s + 4·29-s + 2·37-s + 2·41-s − 4·61-s − 2·73-s − 2·89-s − 2·97-s − 2·109-s + 4·113-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 − 169-s + 173-s + 179-s + 181-s + 4·185-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 14017536 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14017536 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(14017536\)    =    \(2^{10} \cdot 3^{4} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(3.49129\)
Root analytic conductor: \(1.36693\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 14017536,\ (\ :0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.562931291\)
\(L(\frac12)\) \(\approx\) \(2.562931291\)
\(L(1)\) 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 \( 1 \)
3 \( 1 \)
13$C_2$ \( 1 + T^{2} \)
good5$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_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
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$C_1$ \( ( 1 + T )^{4} \)
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

−8.782447344429322762409291767914, −8.763181695814740425332622879866, −8.077196883974705767795534515826, −7.914363884419773255436589195893, −7.28467137467087155827139534592, −7.06536350557419186549458761571, −6.29981781133940215906406851320, −6.28538977692793145722632479705, −6.05006659469721442765779364500, −5.70167595478176793866327755513, −5.08644100637190729889763225675, −4.74554718864563633975788540722, −4.34554960100854779785093159264, −4.15017008628009042565288530091, −2.99740109841387028626714697373, −2.87017453069992817163761397677, −2.64578695447521666299827688073, −2.00928541277879096433162305045, −1.25156413959512857645501907717, −1.12451573249837154139486027186, 1.12451573249837154139486027186, 1.25156413959512857645501907717, 2.00928541277879096433162305045, 2.64578695447521666299827688073, 2.87017453069992817163761397677, 2.99740109841387028626714697373, 4.15017008628009042565288530091, 4.34554960100854779785093159264, 4.74554718864563633975788540722, 5.08644100637190729889763225675, 5.70167595478176793866327755513, 6.05006659469721442765779364500, 6.28538977692793145722632479705, 6.29981781133940215906406851320, 7.06536350557419186549458761571, 7.28467137467087155827139534592, 7.914363884419773255436589195893, 8.077196883974705767795534515826, 8.763181695814740425332622879866, 8.782447344429322762409291767914

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