Properties

Label 4-2e12-1.1-c8e2-0-2
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  + 1.02e3·5-s − 7.03e3·9-s + 5.54e4·13-s + 1.00e5·17-s − 950·25-s − 1.09e5·29-s − 1.58e6·37-s − 1.51e5·41-s − 7.17e6·45-s + 4.99e6·49-s − 2.23e7·53-s + 4.76e7·61-s + 5.65e7·65-s + 1.30e7·73-s + 6.48e6·81-s + 1.02e8·85-s + 1.73e8·89-s − 9.33e7·97-s − 1.31e8·101-s + 3.12e8·109-s + 4.72e8·113-s − 3.90e8·117-s + 6.13e7·121-s − 6.64e8·125-s + 127-s + 131-s + 137-s + ⋯
L(s)  = 1  + 1.63·5-s − 1.07·9-s + 1.94·13-s + 1.20·17-s − 0.00243·25-s − 0.155·29-s − 0.847·37-s − 0.0534·41-s − 1.75·45-s + 0.866·49-s − 2.83·53-s + 3.44·61-s + 3.16·65-s + 0.458·73-s + 0.150·81-s + 1.96·85-s + 2.76·89-s − 1.05·97-s − 1.26·101-s + 2.21·109-s + 2.89·113-s − 2.08·117-s + 0.285·121-s − 2.72·125-s − 0.253·145-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

\(L(\frac{9}{2})\) \(\approx\) \(3.996301550\)
\(L(\frac12)\) \(\approx\) \(3.996301550\)
\(L(5)\) 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$C_2$ \( ( 1 - 26 p T + p^{8} T^{2} )( 1 + 26 p T + p^{8} T^{2} ) \)
5$C_2$ \( ( 1 - 102 p T + p^{8} T^{2} )^{2} \)
7$C_2^2$ \( 1 - 713966 p T^{2} + p^{16} T^{4} \)
11$C_2^2$ \( 1 - 61301762 T^{2} + p^{16} T^{4} \)
13$C_2$ \( ( 1 - 27710 T + p^{8} T^{2} )^{2} \)
17$C_2$ \( ( 1 - 50370 T + p^{8} T^{2} )^{2} \)
19$C_2^2$ \( 1 - 22168990082 T^{2} + p^{16} T^{4} \)
23$C_2^2$ \( 1 - 237285218 p^{2} T^{2} + p^{16} T^{4} \)
29$C_2$ \( ( 1 + 54978 T + p^{8} T^{2} )^{2} \)
31$C_2^2$ \( 1 - 323644730882 T^{2} + p^{16} T^{4} \)
37$C_2$ \( ( 1 + 793730 T + p^{8} T^{2} )^{2} \)
41$C_2$ \( ( 1 + 75582 T + p^{8} T^{2} )^{2} \)
43$C_2^2$ \( 1 - 23126751997442 T^{2} + p^{16} T^{4} \)
47$C_2^2$ \( 1 - 39399744456962 T^{2} + p^{16} T^{4} \)
53$C_2$ \( ( 1 + 11166210 T + p^{8} T^{2} )^{2} \)
59$C_2^2$ \( 1 + 182995617327358 T^{2} + p^{16} T^{4} \)
61$C_2$ \( ( 1 - 23826622 T + p^{8} T^{2} )^{2} \)
67$C_2^2$ \( 1 - 755964429617282 T^{2} + p^{16} T^{4} \)
71$C_2^2$ \( 1 - 1189851873275522 T^{2} + p^{16} T^{4} \)
73$C_2$ \( ( 1 - 6516610 T + p^{8} T^{2} )^{2} \)
79$C_2^2$ \( 1 - 653831579109122 T^{2} + p^{16} T^{4} \)
83$C_2^2$ \( 1 + 890071220357758 T^{2} + p^{16} T^{4} \)
89$C_2$ \( ( 1 - 86795778 T + p^{8} T^{2} )^{2} \)
97$C_2$ \( ( 1 + 46670270 T + p^{8} T^{2} )^{2} \)
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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

−14.16201726287854485747282368923, −13.05945318276671579589730156089, −12.57486145401775053057546387453, −11.65670796818522996740271549094, −11.30574302675563056006778267263, −10.62511150733703880807589528210, −10.02064722060056510810802516437, −9.560503398474940627893706430049, −8.865468779469287258832605893093, −8.386105407345175012495295137309, −7.72374643027032635671405508560, −6.61963186998158943718667266761, −6.04969144757121573360728882367, −5.69903920157139165754757256984, −5.12723086185503322900284643538, −3.77928385935194435663005812495, −3.22086672365923328395314936354, −2.18318653934294933111889175088, −1.54187561082246158699524994837, −0.67580426275643655029046169741, 0.67580426275643655029046169741, 1.54187561082246158699524994837, 2.18318653934294933111889175088, 3.22086672365923328395314936354, 3.77928385935194435663005812495, 5.12723086185503322900284643538, 5.69903920157139165754757256984, 6.04969144757121573360728882367, 6.61963186998158943718667266761, 7.72374643027032635671405508560, 8.386105407345175012495295137309, 8.865468779469287258832605893093, 9.560503398474940627893706430049, 10.02064722060056510810802516437, 10.62511150733703880807589528210, 11.30574302675563056006778267263, 11.65670796818522996740271549094, 12.57486145401775053057546387453, 13.05945318276671579589730156089, 14.16201726287854485747282368923

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