↑

Properties

Label 4-85280-1.1-c1e2-0-11
Degree $4$
Conductor $85280$
Sign $-1$
Analytic cond. $5.43752$
Root an. cond. $1.52703$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s + 8-s − 4·9-s − 8·13-s + 16-s − 6·17-s − 4·18-s − 4·25-s − 8·26-s − 3·29-s + 32-s − 6·34-s − 4·36-s + 6·37-s − 41-s + 8·49-s − 4·50-s − 8·52-s − 11·53-s − 3·58-s − 9·61-s + 64-s − 6·68-s − 4·72-s + 8·73-s + 6·74-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 0.353·8-s − 4/3·9-s − 2.21·13-s + 1/4·16-s − 1.45·17-s − 0.942·18-s − 4/5·25-s − 1.56·26-s − 0.557·29-s + 0.176·32-s − 1.02·34-s − 2/3·36-s + 0.986·37-s − 0.156·41-s + 8/7·49-s − 0.565·50-s − 1.10·52-s − 1.51·53-s − 0.393·58-s − 1.15·61-s + 1/8·64-s − 0.727·68-s − 0.471·72-s + 0.936·73-s + 0.697·74-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 85280 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 85280 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(85280\)    =    \(2^{5} \cdot 5 \cdot 13 \cdot 41\)
Sign: $-1$
Analytic conductor: \(5.43752\)
Root analytic conductor: \(1.52703\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 85280,\ (\ :1/2, 1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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$C_1$ \( 1 - T \)
5$C_1$$\times$$C_2$ \( ( 1 - T )( 1 + T + p T^{2} ) \)
13$C_1$$\times$$C_2$ \( ( 1 + T )( 1 + 7 T + p T^{2} ) \)
41$C_1$$\times$$C_2$ \( ( 1 + T )( 1 + p T^{2} ) \)
good3$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \) 2.3.a_e
7$C_2^2$ \( 1 - 8 T^{2} + p^{2} T^{4} \) 2.7.a_ai
11$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.11.a_ak
17$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.17.g_br
19$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.19.a_aba
23$C_2^2$ \( 1 + 16 T^{2} + p^{2} T^{4} \) 2.23.a_q
29$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.29.d_bw
31$C_2^2$ \( 1 - 24 T^{2} + p^{2} T^{4} \) 2.31.a_ay
37$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.37.ag_bi
43$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.43.a_adi
47$C_2^2$ \( 1 - 40 T^{2} + p^{2} T^{4} \) 2.47.a_abo
53$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.53.l_eu
59$C_2^2$ \( 1 + 11 T^{2} + p^{2} T^{4} \) 2.59.a_l
61$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) 2.61.j_bg
67$C_2^2$ \( 1 - 17 T^{2} + p^{2} T^{4} \) 2.67.a_ar
71$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \) 2.71.a_ba
73$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.73.ai_gg
79$C_2^2$ \( 1 + 16 T^{2} + p^{2} T^{4} \) 2.79.a_q
83$C_2^2$ \( 1 - 114 T^{2} + p^{2} T^{4} \) 2.83.a_aek
89$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.89.c_fa
97$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 - 7 T + p T^{2} ) \) 2.97.av_lg
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

−9.384261794738663379001853040207, −9.063086912699619163012346142942, −8.417820870750715009427410287493, −7.72771612056908536492348139244, −7.50632466772525982110189309162, −6.82463666891408075052796590312, −6.22785631805514179170740754003, −5.81162100815207357523321280126, −5.07886527573010417046111449722, −4.74654933485478492084320673014, −4.09479150991100469310891154811, −3.22362495277057918546523697497, −2.51198349974349672015872256061, −2.11878453942050584428041650988, 0, 2.11878453942050584428041650988, 2.51198349974349672015872256061, 3.22362495277057918546523697497, 4.09479150991100469310891154811, 4.74654933485478492084320673014, 5.07886527573010417046111449722, 5.81162100815207357523321280126, 6.22785631805514179170740754003, 6.82463666891408075052796590312, 7.50632466772525982110189309162, 7.72771612056908536492348139244, 8.417820870750715009427410287493, 9.063086912699619163012346142942, 9.384261794738663379001853040207

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