↑

Properties

Label 4-676000-1.1-c1e2-0-15
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 $0$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s + 5-s + 8-s + 4·9-s + 10-s − 4·13-s + 16-s + 4·17-s + 4·18-s + 20-s + 25-s − 4·26-s + 2·29-s + 32-s + 4·34-s + 4·36-s + 40-s + 8·41-s + 4·45-s − 10·49-s + 50-s − 4·52-s − 10·53-s + 2·58-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 + 4/3·9-s + 0.316·10-s − 1.10·13-s + 1/4·16-s + 0.970·17-s + 0.942·18-s + 0.223·20-s + 1/5·25-s − 0.784·26-s + 0.371·29-s + 0.176·32-s + 0.685·34-s + 2/3·36-s + 0.158·40-s + 1.24·41-s + 0.596·45-s − 1.42·49-s + 0.141·50-s − 0.554·52-s − 1.37·53-s + 0.262·58-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: \(0\)
Selberg data: \((4,\ 676000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.986807596\)
\(L(\frac12)\) \(\approx\) \(3.986807596\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 - 4 T^{2} + p^{2} T^{4} \) 2.3.a_ae
7$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.7.a_k
11$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \) 2.11.a_k
17$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.17.ae_bm
19$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \) 2.19.a_k
23$C_2^2$ \( 1 - 26 T^{2} + p^{2} T^{4} \) 2.23.a_aba
29$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.29.ac_bi
31$C_2^2$ \( 1 - 20 T^{2} + p^{2} T^{4} \) 2.31.a_au
37$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.37.a_cs
41$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.41.ai_ck
43$C_2^2$ \( 1 - 16 T^{2} + p^{2} T^{4} \) 2.43.a_aq
47$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \) 2.47.a_by
53$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.53.k_fa
59$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \) 2.59.a_acs
61$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + p T^{2} ) \) 2.61.ak_es
67$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \) 2.67.a_by
71$C_2^2$ \( 1 + 40 T^{2} + p^{2} T^{4} \) 2.71.a_bo
73$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.73.am_ha
79$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.79.a_agc
83$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) 2.83.a_adm
89$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.89.ae_eo
97$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) 2.97.k_by
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

−8.128905350000375201979656815020, −7.71825097915921139899626005770, −7.48873870745299399308454204049, −6.88956757792854077776028887595, −6.56268894328578737897836046419, −6.08665189710981693573177032003, −5.49830718864943777051207820911, −5.03706339554568022796219852551, −4.65580733523787889671688707417, −4.20421727295777854644490992950, −3.53229178805182855835583123025, −3.04247633435558309219926218555, −2.30622805943280384288205128923, −1.76769095239351648533906960741, −0.936043464981723324417298150710, 0.936043464981723324417298150710, 1.76769095239351648533906960741, 2.30622805943280384288205128923, 3.04247633435558309219926218555, 3.53229178805182855835583123025, 4.20421727295777854644490992950, 4.65580733523787889671688707417, 5.03706339554568022796219852551, 5.49830718864943777051207820911, 6.08665189710981693573177032003, 6.56268894328578737897836046419, 6.88956757792854077776028887595, 7.48873870745299399308454204049, 7.71825097915921139899626005770, 8.128905350000375201979656815020

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