↑

Properties

Label 4-623808-1.1-c1e2-0-16
Degree $4$
Conductor $623808$
Sign $1$
Analytic cond. $39.7745$
Root an. cond. $2.51131$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 3-s − 4-s + 6-s + 3·8-s + 9-s + 12-s − 16-s − 18-s + 4·19-s − 3·24-s − 2·25-s − 27-s + 8·29-s − 5·32-s − 36-s − 4·38-s + 4·41-s + 8·43-s + 48-s − 14·49-s + 2·50-s + 16·53-s + 54-s − 4·57-s − 8·58-s − 8·61-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s − 1/2·4-s + 0.408·6-s + 1.06·8-s + 1/3·9-s + 0.288·12-s − 1/4·16-s − 0.235·18-s + 0.917·19-s − 0.612·24-s − 2/5·25-s − 0.192·27-s + 1.48·29-s − 0.883·32-s − 1/6·36-s − 0.648·38-s + 0.624·41-s + 1.21·43-s + 0.144·48-s − 2·49-s + 0.282·50-s + 2.19·53-s + 0.136·54-s − 0.529·57-s − 1.05·58-s − 1.02·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 623808 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 623808 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(623808\)    =    \(2^{6} \cdot 3^{3} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(39.7745\)
Root analytic conductor: \(2.51131\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 623808,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8812240216\)
\(L(\frac12)\) \(\approx\) \(0.8812240216\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T + p T^{2} \)
3$C_1$ \( 1 + T \)
19$C_2$ \( 1 - 4 T + p T^{2} \)
good5$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.5.a_c
7$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.7.a_o
11$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.11.a_ak
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.13.a_k
17$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \) 2.17.a_as
23$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \) 2.23.a_aw
29$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.29.ai_bm
31$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \) 2.31.a_by
37$C_2^2$ \( 1 - 54 T^{2} + p^{2} T^{4} \) 2.37.a_acc
41$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.41.ae_cs
43$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.43.ai_bm
47$C_2^2$ \( 1 - 54 T^{2} + p^{2} T^{4} \) 2.47.a_acc
53$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) 2.53.aq_gk
59$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.59.a_dy
61$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.61.i_fe
67$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.67.a_ak
71$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) 2.71.ai_fm
73$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.73.e_di
79$C_2^2$ \( 1 - 62 T^{2} + p^{2} T^{4} \) 2.79.a_ack
83$C_2^2$ \( 1 + 86 T^{2} + p^{2} T^{4} \) 2.83.a_di
89$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.89.ae_eo
97$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.97.a_gc
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

−8.446506929753249545177900870577, −7.974343655440478531711444894748, −7.51807052948735244360197008663, −7.19139755277043271422456892253, −6.65114644039420565670907404565, −6.08028370479278811397695239367, −5.66472679810901525759857455071, −5.11663810477219539052022762572, −4.66406356614351487976824993967, −4.24725165718132712064708129082, −3.62478510158769429706429958280, −2.96508949926411653841300894251, −2.16241729124266777505565848848, −1.29162763419450906864019887941, −0.63239715124165236793703764842, 0.63239715124165236793703764842, 1.29162763419450906864019887941, 2.16241729124266777505565848848, 2.96508949926411653841300894251, 3.62478510158769429706429958280, 4.24725165718132712064708129082, 4.66406356614351487976824993967, 5.11663810477219539052022762572, 5.66472679810901525759857455071, 6.08028370479278811397695239367, 6.65114644039420565670907404565, 7.19139755277043271422456892253, 7.51807052948735244360197008663, 7.974343655440478531711444894748, 8.446506929753249545177900870577

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