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Properties

Label 4-2e8-1.1-c18e2-0-0
Degree $4$
Conductor $256$
Sign $1$
Analytic cond. $1079.89$
Root an. cond. $5.73251$
Motivic weight $18$
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  − 5.58e6·5-s − 1.99e8·9-s + 2.60e10·13-s − 1.91e11·17-s + 1.58e13·25-s − 2.40e13·29-s + 1.60e14·37-s + 7.93e14·41-s + 1.11e15·45-s + 3.17e15·49-s − 5.96e14·53-s + 1.62e16·61-s − 1.45e17·65-s + 1.60e17·73-s − 1.10e17·81-s + 1.07e18·85-s + 2.67e17·89-s + 1.19e18·97-s − 2.14e18·101-s − 4.69e18·109-s + 7.39e18·113-s − 5.19e18·117-s − 5.48e18·121-s − 2.33e19·125-s + 127-s + 131-s + 137-s + ⋯
L(s)  = 1  − 2.86·5-s − 0.514·9-s + 2.45·13-s − 1.61·17-s + 4.14·25-s − 1.65·29-s + 1.23·37-s + 2.42·41-s + 1.47·45-s + 1.95·49-s − 0.180·53-s + 1.38·61-s − 7.02·65-s + 2.72·73-s − 0.735·81-s + 4.63·85-s + 0.763·89-s + 1.57·97-s − 1.96·101-s − 2.16·109-s + 2.46·113-s − 1.26·117-s − 0.986·121-s − 3.13·125-s + 4.74·145-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(19-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+9)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(256\)    =    \(2^{8}\)
Sign: $1$
Analytic conductor: \(1079.89\)
Root analytic conductor: \(5.73251\)
Motivic weight: \(18\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 256,\ (\ :9, 9),\ 1)\)

Particular Values

\(L(\frac{19}{2})\) \(\approx\) \(1.116269594\)
\(L(\frac12)\) \(\approx\) \(1.116269594\)
\(L(10)\) 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$C_2^2$ \( 1 + 7384106 p^{3} T^{2} + p^{36} T^{4} \)
5$C_2$ \( ( 1 + 111798 p^{2} T + p^{18} T^{2} )^{2} \)
7$C_2^2$ \( 1 - 454193424779534 p T^{2} + p^{36} T^{4} \)
11$C_2^2$ \( 1 + 5482680652015313038 T^{2} + p^{36} T^{4} \)
13$C_2$ \( ( 1 - 1001418610 p T + p^{18} T^{2} )^{2} \)
17$C_2$ \( ( 1 + 5644983870 p T + p^{18} T^{2} )^{2} \)
19$C_2^2$ \( 1 - \)\(17\!\cdots\!82\)\( T^{2} + p^{36} T^{4} \)
23$C_2^2$ \( 1 + \)\(32\!\cdots\!22\)\( T^{2} + p^{36} T^{4} \)
29$C_2$ \( ( 1 + 12018060640278 T + p^{18} T^{2} )^{2} \)
31$C_2^2$ \( 1 - \)\(31\!\cdots\!82\)\( T^{2} + p^{36} T^{4} \)
37$C_2$ \( ( 1 - 80364756548090 T + p^{18} T^{2} )^{2} \)
41$C_2$ \( ( 1 - 396598947984018 T + p^{18} T^{2} )^{2} \)
43$C_2^2$ \( 1 - \)\(15\!\cdots\!58\)\( T^{2} + p^{36} T^{4} \)
47$C_2^2$ \( 1 - \)\(23\!\cdots\!38\)\( T^{2} + p^{36} T^{4} \)
53$C_2$ \( ( 1 + 298022493539430 T + p^{18} T^{2} )^{2} \)
59$C_2^2$ \( 1 - \)\(13\!\cdots\!42\)\( T^{2} + p^{36} T^{4} \)
61$C_2$ \( ( 1 - 8111952901924778 T + p^{18} T^{2} )^{2} \)
67$C_2^2$ \( 1 + \)\(12\!\cdots\!82\)\( T^{2} + p^{36} T^{4} \)
71$C_2^2$ \( 1 - \)\(42\!\cdots\!22\)\( T^{2} + p^{36} T^{4} \)
73$C_2$ \( ( 1 - 80276288606916370 T + p^{18} T^{2} )^{2} \)
79$C_2^2$ \( 1 - \)\(28\!\cdots\!22\)\( T^{2} + p^{36} T^{4} \)
83$C_2^2$ \( 1 - \)\(68\!\cdots\!58\)\( T^{2} + p^{36} T^{4} \)
89$C_2$ \( ( 1 - 133741369608374322 T + p^{18} T^{2} )^{2} \)
97$C_2$ \( ( 1 - 598783793423497090 T + p^{18} T^{2} )^{2} \)
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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

−15.25929278433007025466265057720, −14.72843217547095467412201216541, −13.65387825010726616340018561645, −13.01412337163858138296277823547, −12.27955393792363503738762063010, −11.36573238942922544400990047695, −11.13040988605784989314433112557, −10.96159934851692812426104211441, −9.151163554481969562731019239364, −8.640494995836707055812749404318, −8.015682236367187503475091339857, −7.50236861366441622813517657396, −6.58599834850796927454719708620, −5.71596202990603170359142867529, −4.32888985894059499597532798279, −3.93941985729555776886042174341, −3.54733117020627222367294650923, −2.39492732052076005971357283361, −0.928780205097809786459672923812, −0.43478565331976052623555922326, 0.43478565331976052623555922326, 0.928780205097809786459672923812, 2.39492732052076005971357283361, 3.54733117020627222367294650923, 3.93941985729555776886042174341, 4.32888985894059499597532798279, 5.71596202990603170359142867529, 6.58599834850796927454719708620, 7.50236861366441622813517657396, 8.015682236367187503475091339857, 8.640494995836707055812749404318, 9.151163554481969562731019239364, 10.96159934851692812426104211441, 11.13040988605784989314433112557, 11.36573238942922544400990047695, 12.27955393792363503738762063010, 13.01412337163858138296277823547, 13.65387825010726616340018561645, 14.72843217547095467412201216541, 15.25929278433007025466265057720

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