↑

Properties

Label 4-24e4-1.1-c1e2-0-19
Degree $4$
Conductor $331776$
Sign $1$
Analytic cond. $21.1543$
Root an. cond. $2.14461$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 8·13-s + 8·25-s − 4·37-s + 14·49-s + 20·61-s − 32·73-s + 16·97-s − 40·109-s − 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 22·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯
L(s)  = 1  + 2.21·13-s + 8/5·25-s − 0.657·37-s + 2·49-s + 2.56·61-s − 3.74·73-s + 1.62·97-s − 3.83·109-s − 2·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 1.69·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 331776 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 331776 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(331776\)    =    \(2^{12} \cdot 3^{4}\)
Sign: $1$
Analytic conductor: \(21.1543\)
Root analytic conductor: \(2.14461\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 331776,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.132693776\)
\(L(\frac12)\) \(\approx\) \(2.132693776\)
\(L(\frac{3}{2})\) 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)$Isogeny Class over $\mathbf{F}_p$
bad2 \( 1 \)
3 \( 1 \)
good5$C_2^2$ \( 1 - 8 T^{2} + p^{2} T^{4} \) 2.5.a_ai
7$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.7.a_ao
11$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.11.a_w
13$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.13.ai_bq
17$C_2^2$ \( 1 + 16 T^{2} + p^{2} T^{4} \) 2.17.a_q
19$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.19.a_abm
23$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.23.a_bu
29$C_2^2$ \( 1 + 40 T^{2} + p^{2} T^{4} \) 2.29.a_bo
31$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.31.a_ack
37$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.37.e_da
41$C_2^2$ \( 1 - 80 T^{2} + p^{2} T^{4} \) 2.41.a_adc
43$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.43.a_adi
47$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.47.a_dq
53$C_2^2$ \( 1 - 56 T^{2} + p^{2} T^{4} \) 2.53.a_ace
59$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.59.a_eo
61$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.61.au_io
67$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.67.a_afe
71$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.71.a_fm
73$C_2$ \( ( 1 + 16 T + p T^{2} )^{2} \) 2.73.bg_pm
79$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.79.a_agc
83$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.83.a_gk
89$C_2^2$ \( 1 + 160 T^{2} + p^{2} T^{4} \) 2.89.a_ge
97$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.97.aq_jy
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.68275052586659087921319497973, −10.61885454202765275179954157864, −10.31670034381543091886270597365, −9.575390039087647071437045154020, −8.915288129991793242359600146610, −8.892310306806202347208647108774, −8.365345937509952469990148640623, −8.022253874225529949051092210267, −7.15726967605187408569971082822, −7.01748190609164655772193904811, −6.37892264030131945486383552163, −5.95231434978599084789235368728, −5.45561806442231171703791859695, −4.98753274533784914708833691558, −4.04403104942718968977940689424, −3.97991842024068186800107769066, −3.15271126923595400466103256639, −2.62048424389790237137731620764, −1.59127865434395909682260622352, −0.927493852611657402345888739097, 0.927493852611657402345888739097, 1.59127865434395909682260622352, 2.62048424389790237137731620764, 3.15271126923595400466103256639, 3.97991842024068186800107769066, 4.04403104942718968977940689424, 4.98753274533784914708833691558, 5.45561806442231171703791859695, 5.95231434978599084789235368728, 6.37892264030131945486383552163, 7.01748190609164655772193904811, 7.15726967605187408569971082822, 8.022253874225529949051092210267, 8.365345937509952469990148640623, 8.892310306806202347208647108774, 8.915288129991793242359600146610, 9.575390039087647071437045154020, 10.31670034381543091886270597365, 10.61885454202765275179954157864, 10.68275052586659087921319497973

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