Properties

Degree $4$
Conductor $65536$
Sign $1$
Motivic weight $5$
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 416·7-s + 422·9-s − 2.55e3·17-s + 6.43e3·23-s + 6.05e3·25-s + 5.24e3·31-s − 340·41-s − 64·47-s + 9.61e4·49-s − 1.75e5·63-s + 5.71e4·71-s + 1.07e5·73-s + 1.38e5·79-s + 1.19e5·81-s + 2.53e5·89-s + 1.24e5·97-s − 2.16e5·103-s + 3.15e4·113-s + 1.06e6·119-s + 3.48e4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 1.07e6·153-s + ⋯
L(s)  = 1  − 3.20·7-s + 1.73·9-s − 2.14·17-s + 2.53·23-s + 1.93·25-s + 0.980·31-s − 0.0315·41-s − 0.00422·47-s + 5.72·49-s − 5.57·63-s + 1.34·71-s + 2.35·73-s + 2.49·79-s + 2.01·81-s + 3.39·89-s + 1.34·97-s − 2.01·103-s + 0.232·113-s + 6.88·119-s + 0.216·121-s + 5.50e−6·127-s + 5.09e−6·131-s + 4.55e−6·137-s + 4.38e−6·139-s + 3.69e−6·149-s + 3.56e−6·151-s − 3.72·153-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 65536 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65536 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(65536\)    =    \(2^{16}\)
Sign: $1$
Motivic weight: \(5\)
Character: induced by $\chi_{256} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 65536,\ (\ :5/2, 5/2),\ 1)\)

Particular Values

\(L(3)\) \(\approx\) \(2.349282525\)
\(L(\frac12)\) \(\approx\) \(2.349282525\)
\(L(\frac{7}{2})\) 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 \( 1 \)
good3$C_2^2$ \( 1 - 422 T^{2} + p^{10} T^{4} \)
5$C_2^2$ \( 1 - 6054 T^{2} + p^{10} T^{4} \)
7$C_2$ \( ( 1 + 208 T + p^{5} T^{2} )^{2} \)
11$C_2^2$ \( 1 - 34806 T^{2} + p^{10} T^{4} \)
13$C_2^2$ \( 1 - 260950 T^{2} + p^{10} T^{4} \)
17$C_2$ \( ( 1 + 1278 T + p^{5} T^{2} )^{2} \)
19$C_2^2$ \( 1 - 3715654 T^{2} + p^{10} T^{4} \)
23$C_2$ \( ( 1 - 3216 T + p^{5} T^{2} )^{2} \)
29$C_2^2$ \( 1 - 32507574 T^{2} + p^{10} T^{4} \)
31$C_2$ \( ( 1 - 2624 T + p^{5} T^{2} )^{2} \)
37$C_2^2$ \( 1 - 49234150 T^{2} + p^{10} T^{4} \)
41$C_2$ \( ( 1 + 170 T + p^{5} T^{2} )^{2} \)
43$C_2^2$ \( 1 + 103108298 T^{2} + p^{10} T^{4} \)
47$C_2$ \( ( 1 + 32 T + p^{5} T^{2} )^{2} \)
53$C_2^2$ \( 1 - 344527302 T^{2} + p^{10} T^{4} \)
59$C_2^2$ \( 1 + 290741802 T^{2} + p^{10} T^{4} \)
61$C_2^2$ \( 1 - 1450119158 T^{2} + p^{10} T^{4} \)
67$C_2^2$ \( 1 - 2269936678 T^{2} + p^{10} T^{4} \)
71$C_2$ \( ( 1 - 28592 T + p^{5} T^{2} )^{2} \)
73$C_2$ \( ( 1 - 53670 T + p^{5} T^{2} )^{2} \)
79$C_2$ \( ( 1 - 69152 T + p^{5} T^{2} )^{2} \)
83$C_2^2$ \( 1 - 6449241286 T^{2} + p^{10} T^{4} \)
89$C_2$ \( ( 1 - 126806 T + p^{5} T^{2} )^{2} \)
97$C_2$ \( ( 1 - 62290 T + p^{5} T^{2} )^{2} \)
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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

−11.09851209440181866137820947320, −10.89628636939291445074987756995, −10.33429823638872937654033202665, −9.976498169093567598215292574664, −9.371347617413296120511070335792, −9.069069964767140883089591752317, −8.963908952979392172757969069897, −7.88130016279287376779156749749, −6.97465308288673453044153094673, −6.90115224621128176052192026829, −6.46851200471853872958380150271, −6.36974902341444197604499602974, −4.99245455712711890759791353660, −4.79850606583301780160859740845, −3.86253088283351843367926703351, −3.43552373123927133286410015725, −2.81658386010271444146530465837, −2.24198800218706059853707855321, −0.853137605783157883804759930761, −0.60220164904891101494303350877, 0.60220164904891101494303350877, 0.853137605783157883804759930761, 2.24198800218706059853707855321, 2.81658386010271444146530465837, 3.43552373123927133286410015725, 3.86253088283351843367926703351, 4.79850606583301780160859740845, 4.99245455712711890759791353660, 6.36974902341444197604499602974, 6.46851200471853872958380150271, 6.90115224621128176052192026829, 6.97465308288673453044153094673, 7.88130016279287376779156749749, 8.963908952979392172757969069897, 9.069069964767140883089591752317, 9.371347617413296120511070335792, 9.976498169093567598215292574664, 10.33429823638872937654033202665, 10.89628636939291445074987756995, 11.09851209440181866137820947320

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