↑

Properties

Label 4-676000-1.1-c1e2-0-38
Degree $4$
Conductor $676000$
Sign $-1$
Analytic cond. $43.1023$
Root an. cond. $2.56227$
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 + 5-s + 8-s − 9-s + 10-s − 4·13-s + 16-s − 17-s − 18-s + 20-s + 25-s − 4·26-s − 3·29-s + 32-s − 34-s − 36-s − 5·37-s + 40-s − 17·41-s − 45-s + 5·49-s + 50-s − 4·52-s − 3·58-s − 10·61-s + 64-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 0.447·5-s + 0.353·8-s − 1/3·9-s + 0.316·10-s − 1.10·13-s + 1/4·16-s − 0.242·17-s − 0.235·18-s + 0.223·20-s + 1/5·25-s − 0.784·26-s − 0.557·29-s + 0.176·32-s − 0.171·34-s − 1/6·36-s − 0.821·37-s + 0.158·40-s − 2.65·41-s − 0.149·45-s + 5/7·49-s + 0.141·50-s − 0.554·52-s − 0.393·58-s − 1.28·61-s + 1/8·64-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 676000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 676000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(676000\)    =    \(2^{5} \cdot 5^{3} \cdot 13^{2}\)
Sign: $-1$
Analytic conductor: \(43.1023\)
Root analytic conductor: \(2.56227\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 676000,\ (\ :1/2, 1/2),\ -1)\)

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$C_1$ \( 1 - T \)
13$C_2$ \( 1 + 4 T + p T^{2} \)
good3$C_2^2$ \( 1 + T^{2} + p^{2} T^{4} \) 2.3.a_b
7$C_2^2$ \( 1 - 5 T^{2} + p^{2} T^{4} \) 2.7.a_af
11$C_2^2$ \( 1 - 5 T^{2} + p^{2} T^{4} \) 2.11.a_af
17$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.17.b_bc
19$C_2^2$ \( 1 - 35 T^{2} + p^{2} T^{4} \) 2.19.a_abj
23$C_2^2$ \( 1 + 34 T^{2} + p^{2} T^{4} \) 2.23.a_bi
29$C_2$$\times$$C_2$ \( ( 1 - T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.29.d_cc
31$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \) 2.31.a_abe
37$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.37.f_by
41$C_2$$\times$$C_2$ \( ( 1 + 7 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.41.r_fw
43$C_2^2$ \( 1 + 69 T^{2} + p^{2} T^{4} \) 2.43.a_cr
47$C_2^2$ \( 1 + 5 T^{2} + p^{2} T^{4} \) 2.47.a_f
53$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.53.a_z
59$C_2^2$ \( 1 + 25 T^{2} + p^{2} T^{4} \) 2.59.a_z
61$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.61.k_es
67$C_2^2$ \( 1 - 100 T^{2} + p^{2} T^{4} \) 2.67.a_adw
71$C_2^2$ \( 1 - 40 T^{2} + p^{2} T^{4} \) 2.71.a_abo
73$C_2$$\times$$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.73.am_de
79$C_2^2$ \( 1 + 67 T^{2} + p^{2} T^{4} \) 2.79.a_cp
83$C_2^2$ \( 1 - 40 T^{2} + p^{2} T^{4} \) 2.83.a_abo
89$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.89.g_gw
97$C_2$$\times$$C_2$ \( ( 1 - 17 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) 2.97.au_jl
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

−8.082472094627941710093637812080, −7.55578208164520834487163091993, −7.13084724994494408888935609576, −6.71175815640968119936341876103, −6.34159010508336811317256347955, −5.68974781626705348852806507018, −5.34272554334946784137713968118, −4.89302884058171014734493601292, −4.51442291146561926500165057493, −3.70893207038840567270180428518, −3.33738663664989608525632499688, −2.63158096804588805240594955181, −2.12590674464019278469470476240, −1.45387958101856739829733637647, 0, 1.45387958101856739829733637647, 2.12590674464019278469470476240, 2.63158096804588805240594955181, 3.33738663664989608525632499688, 3.70893207038840567270180428518, 4.51442291146561926500165057493, 4.89302884058171014734493601292, 5.34272554334946784137713968118, 5.68974781626705348852806507018, 6.34159010508336811317256347955, 6.71175815640968119936341876103, 7.13084724994494408888935609576, 7.55578208164520834487163091993, 8.082472094627941710093637812080

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