Properties

Label 4-2e12-1.1-c8e2-0-0
Degree $4$
Conductor $4096$
Sign $1$
Analytic cond. $679.761$
Root an. cond. $5.10609$
Motivic weight $8$
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  − 516·5-s + 1.29e4·9-s − 3.82e4·13-s − 1.17e5·17-s − 5.81e5·25-s − 1.68e6·29-s − 5.09e6·37-s − 8.64e6·41-s − 6.67e6·45-s + 6.53e5·49-s − 2.38e6·53-s − 1.68e7·61-s + 1.97e7·65-s + 2.54e7·73-s + 1.24e8·81-s + 6.05e7·85-s − 3.36e7·89-s + 2.41e8·97-s − 2.32e7·101-s − 3.11e7·109-s − 9.28e7·113-s − 4.94e8·117-s − 1.07e8·121-s + 5.35e8·125-s + 127-s + 131-s + 137-s + ⋯
L(s)  = 1  − 0.825·5-s + 1.97·9-s − 1.34·13-s − 1.40·17-s − 1.48·25-s − 2.38·29-s − 2.71·37-s − 3.06·41-s − 1.62·45-s + 0.113·49-s − 0.302·53-s − 1.21·61-s + 1.10·65-s + 0.896·73-s + 2.88·81-s + 1.16·85-s − 0.535·89-s + 2.73·97-s − 0.223·101-s − 0.220·109-s − 0.569·113-s − 2.64·117-s − 0.500·121-s + 2.19·125-s + 1.96·145-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4096 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4096 ^{s/2} \, \Gamma_{\C}(s+4)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(4096\)    =    \(2^{12}\)
Sign: $1$
Analytic conductor: \(679.761\)
Root analytic conductor: \(5.10609\)
Motivic weight: \(8\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 4096,\ (\ :4, 4),\ 1)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(0.005614785689\)
\(L(\frac12)\) \(\approx\) \(0.005614785689\)
\(L(5)\) 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$C_2^2$ \( 1 - 4310 p T^{2} + p^{16} T^{4} \)
5$C_2$ \( ( 1 + 258 T + p^{8} T^{2} )^{2} \)
7$C_2^2$ \( 1 - 13346 p^{2} T^{2} + p^{16} T^{4} \)
11$C_2^2$ \( 1 + 107392510 T^{2} + p^{16} T^{4} \)
13$C_2$ \( ( 1 + 19138 T + p^{8} T^{2} )^{2} \)
17$C_2$ \( ( 1 + 58686 T + p^{8} T^{2} )^{2} \)
19$C_2^2$ \( 1 - 10688638850 T^{2} + p^{16} T^{4} \)
23$C_2^2$ \( 1 - 83658891650 T^{2} + p^{16} T^{4} \)
29$C_2$ \( ( 1 + 842178 T + p^{8} T^{2} )^{2} \)
31$C_2^2$ \( 1 - 600755547650 T^{2} + p^{16} T^{4} \)
37$C_2$ \( ( 1 + 2548610 T + p^{8} T^{2} )^{2} \)
41$C_2$ \( ( 1 + 4324158 T + p^{8} T^{2} )^{2} \)
43$C_2^2$ \( 1 - 19226964507650 T^{2} + p^{16} T^{4} \)
47$C_2^2$ \( 1 + 4696760080126 T^{2} + p^{16} T^{4} \)
53$C_2$ \( ( 1 + 1192194 T + p^{8} T^{2} )^{2} \)
59$C_2^2$ \( 1 - 293546462999810 T^{2} + p^{16} T^{4} \)
61$C_2$ \( ( 1 + 8414786 T + p^{8} T^{2} )^{2} \)
67$C_2^2$ \( 1 - 508514414548610 T^{2} + p^{16} T^{4} \)
71$C_2^2$ \( 1 - 338146626840194 T^{2} + p^{16} T^{4} \)
73$C_2$ \( ( 1 - 12735874 T + p^{8} T^{2} )^{2} \)
79$C_2^2$ \( 1 - 2994695631333122 T^{2} + p^{16} T^{4} \)
83$C_2^2$ \( 1 + 2396699180600446 T^{2} + p^{16} T^{4} \)
89$C_2$ \( ( 1 + 16802814 T + p^{8} T^{2} )^{2} \)
97$C_2$ \( ( 1 - 120994882 T + p^{8} T^{2} )^{2} \)
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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

−13.73597572633093835273293092661, −12.80531694665068961350534995158, −12.50186142433538218359023207131, −11.76423351061629709805271468541, −11.49111458452911547523134808675, −10.43130329267950802200397165154, −10.26048025035458753923483003601, −9.475327504056648055887886906184, −9.002071808455315607576723112833, −8.055914698846520058253841669619, −7.44031740412603714780588492395, −7.09804375009977861502747973590, −6.51894939011918220630942065720, −5.25535252139437073084094850646, −4.75388920530747124927762780763, −3.93129292890014117651508133901, −3.53026232875819314771272358087, −1.91813189883442363503805502673, −1.78446947195268763178087946127, −0.02343605033965435338904396661, 0.02343605033965435338904396661, 1.78446947195268763178087946127, 1.91813189883442363503805502673, 3.53026232875819314771272358087, 3.93129292890014117651508133901, 4.75388920530747124927762780763, 5.25535252139437073084094850646, 6.51894939011918220630942065720, 7.09804375009977861502747973590, 7.44031740412603714780588492395, 8.055914698846520058253841669619, 9.002071808455315607576723112833, 9.475327504056648055887886906184, 10.26048025035458753923483003601, 10.43130329267950802200397165154, 11.49111458452911547523134808675, 11.76423351061629709805271468541, 12.50186142433538218359023207131, 12.80531694665068961350534995158, 13.73597572633093835273293092661

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