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Properties

Label 4-2e8-1.1-c16e2-0-0
Degree $4$
Conductor $256$
Sign $1$
Analytic cond. $674.540$
Root an. cond. $5.09626$
Motivic weight $16$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 60·5-s + 4.28e7·9-s − 7.41e7·13-s − 2.76e9·17-s − 3.05e11·25-s − 1.44e12·29-s − 1.00e13·37-s − 1.32e13·41-s − 2.57e9·45-s + 5.72e13·49-s + 1.48e14·53-s + 3.00e14·61-s + 4.45e9·65-s − 1.80e15·73-s − 1.68e13·81-s + 1.65e11·85-s − 3.07e14·89-s + 7.85e15·97-s + 3.08e16·101-s + 2.98e16·109-s − 5.38e16·113-s − 3.17e15·117-s − 3.14e16·121-s + 2.74e13·125-s + 127-s + 131-s + 137-s + ⋯
L(s)  = 1  − 0.000153·5-s + 0.995·9-s − 0.0909·13-s − 0.396·17-s − 1.99·25-s − 2.89·29-s − 2.85·37-s − 1.66·41-s − 0.000152·45-s + 1.72·49-s + 2.38·53-s + 1.56·61-s + 1.39e−5·65-s − 2.23·73-s − 0.00910·81-s + 6.09e−5·85-s − 0.0781·89-s + 1.00·97-s + 2.84·101-s + 1.50·109-s − 2.02·113-s − 0.0905·117-s − 0.685·121-s + 0.000460·125-s + 0.000443·145-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(17-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+8)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(256\)    =    \(2^{8}\)
Sign: $1$
Analytic conductor: \(674.540\)
Root analytic conductor: \(5.09626\)
Motivic weight: \(16\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 256,\ (\ :8, 8),\ 1)\)

Particular Values

\(L(\frac{17}{2})\) \(\approx\) \(1.328075409\)
\(L(\frac12)\) \(\approx\) \(1.328075409\)
\(L(9)\) 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$C_2^2$ \( 1 - 1587046 p^{3} T^{2} + p^{32} T^{4} \)
5$C_2$ \( ( 1 + 6 p T + p^{16} T^{2} )^{2} \)
7$C_2^2$ \( 1 - 166868395214 p^{3} T^{2} + p^{32} T^{4} \)
11$C_2^2$ \( 1 + 2861855000088698 p T^{2} + p^{32} T^{4} \)
13$C_2$ \( ( 1 + 2852630 p T + p^{16} T^{2} )^{2} \)
17$C_2$ \( ( 1 + 1382996670 T + p^{16} T^{2} )^{2} \)
19$C_2^2$ \( 1 - \)\(22\!\cdots\!62\)\( T^{2} + p^{32} T^{4} \)
23$C_2^2$ \( 1 - 20802158013091816418 p^{2} T^{2} + p^{32} T^{4} \)
29$C_2$ \( ( 1 + 722997605022 T + p^{16} T^{2} )^{2} \)
31$C_2^2$ \( 1 - \)\(53\!\cdots\!62\)\( T^{2} + p^{32} T^{4} \)
37$C_2$ \( ( 1 + 5014816412830 T + p^{16} T^{2} )^{2} \)
41$C_2$ \( ( 1 + 6629976268542 T + p^{16} T^{2} )^{2} \)
43$C_2^2$ \( 1 + \)\(58\!\cdots\!98\)\( T^{2} + p^{32} T^{4} \)
47$C_2^2$ \( 1 - \)\(15\!\cdots\!42\)\( T^{2} + p^{32} T^{4} \)
53$C_2$ \( ( 1 - 74221986126690 T + p^{16} T^{2} )^{2} \)
59$C_2^2$ \( 1 - \)\(21\!\cdots\!82\)\( T^{2} + p^{32} T^{4} \)
61$C_2$ \( ( 1 - 150205654477538 T + p^{16} T^{2} )^{2} \)
67$C_2^2$ \( 1 - \)\(19\!\cdots\!62\)\( T^{2} + p^{32} T^{4} \)
71$C_2^2$ \( 1 - \)\(79\!\cdots\!42\)\( T^{2} + p^{32} T^{4} \)
73$C_2$ \( ( 1 + 900413927701310 T + p^{16} T^{2} )^{2} \)
79$C_2^2$ \( 1 + \)\(24\!\cdots\!38\)\( p^{2} T^{2} + p^{32} T^{4} \)
83$C_2^2$ \( 1 - \)\(87\!\cdots\!62\)\( T^{2} + p^{32} T^{4} \)
89$C_2$ \( ( 1 + 153808569932862 T + p^{16} T^{2} )^{2} \)
97$C_2$ \( ( 1 - 3929332845647170 T + p^{16} T^{2} )^{2} \)
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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

−15.50153149566078775638053167789, −15.06293669114051817047736626942, −14.13854556392933183354938012542, −13.26979188700875189298702040780, −13.13639604056306844153076350095, −11.92772164229502295325465816373, −11.65919172422358380889521545202, −10.46618805797197050109592942799, −10.10096563781857494272426840687, −9.181070392882523533974209433632, −8.508269602784004826579472760536, −7.22354813260982874680780327548, −7.22260835595821149102015580566, −5.83864791205803209949562826974, −5.21898398012927630478983235484, −3.99311556059548119340931006080, −3.62755547755021786915466234972, −2.07152212909942746605490769933, −1.70011107795460259911698036419, −0.36880710560429909518556266281, 0.36880710560429909518556266281, 1.70011107795460259911698036419, 2.07152212909942746605490769933, 3.62755547755021786915466234972, 3.99311556059548119340931006080, 5.21898398012927630478983235484, 5.83864791205803209949562826974, 7.22260835595821149102015580566, 7.22354813260982874680780327548, 8.508269602784004826579472760536, 9.181070392882523533974209433632, 10.10096563781857494272426840687, 10.46618805797197050109592942799, 11.65919172422358380889521545202, 11.92772164229502295325465816373, 13.13639604056306844153076350095, 13.26979188700875189298702040780, 14.13854556392933183354938012542, 15.06293669114051817047736626942, 15.50153149566078775638053167789

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