Properties

Label 4-676000-1.1-c1e2-0-1
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 − 9-s − 10-s − 5·13-s + 16-s − 9·17-s + 18-s + 20-s + 25-s + 5·26-s + 15·29-s − 32-s + 9·34-s − 36-s − 14·37-s − 40-s + 6·41-s − 45-s − 4·49-s − 50-s − 5·52-s + 18·53-s − 15·58-s − 13·61-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 − 1.38·13-s + 1/4·16-s − 2.18·17-s + 0.235·18-s + 0.223·20-s + 1/5·25-s + 0.980·26-s + 2.78·29-s − 0.176·32-s + 1.54·34-s − 1/6·36-s − 2.30·37-s − 0.158·40-s + 0.937·41-s − 0.149·45-s − 4/7·49-s − 0.141·50-s − 0.693·52-s + 2.47·53-s − 1.96·58-s − 1.66·61-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

\(L(1)\) \(\approx\) \(0.9076787239\)
\(L(\frac12)\) \(\approx\) \(0.9076787239\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + 5 T + p T^{2} \)
good3$C_2^2$ \( 1 + T^{2} + p^{2} T^{4} \) 2.3.a_b
7$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \) 2.7.a_e
11$C_2^2$ \( 1 + 8 T^{2} + p^{2} T^{4} \) 2.11.a_i
17$C_2$$\times$$C_2$ \( ( 1 + 3 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.17.j_ca
19$C_2^2$ \( 1 - 16 T^{2} + p^{2} T^{4} \) 2.19.a_aq
23$C_2^2$ \( 1 + 20 T^{2} + p^{2} T^{4} \) 2.23.a_u
29$C_2$$\times$$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) 2.29.ap_ei
31$C_2^2$ \( 1 + 5 T^{2} + p^{2} T^{4} \) 2.31.a_f
37$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.37.o_ek
41$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) 2.41.ag_de
43$C_2^2$ \( 1 + 38 T^{2} + p^{2} T^{4} \) 2.43.a_bm
47$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \) 2.47.a_e
53$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \) 2.53.as_hf
59$C_2^2$ \( 1 - 28 T^{2} + p^{2} T^{4} \) 2.59.a_abc
61$C_2$$\times$$C_2$ \( ( 1 + 5 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.61.n_gg
67$C_2^2$ \( 1 + 61 T^{2} + p^{2} T^{4} \) 2.67.a_cj
71$C_2^2$ \( 1 + 11 T^{2} + p^{2} T^{4} \) 2.71.a_l
73$C_2$$\times$$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) 2.73.an_gm
79$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) 2.79.a_adu
83$C_2^2$ \( 1 - 137 T^{2} + p^{2} T^{4} \) 2.83.a_afh
89$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.89.m_gw
97$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.97.ai_gs
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

−8.529988165970825784152675001891, −8.032007711018015513270642474185, −7.37665998210839182118262725385, −6.89767908990416466440617658362, −6.74623413736440975242841230138, −6.26975174528809199200487816086, −5.62334711197677024665142943185, −5.12025243774361693168193973260, −4.60597797100542210688415800585, −4.26833957638228030276239290218, −3.33185571109910878864208535461, −2.61614549789395435360235983313, −2.40018204674207451190242115926, −1.65755483092687820186304136935, −0.51235783966237159036083617522, 0.51235783966237159036083617522, 1.65755483092687820186304136935, 2.40018204674207451190242115926, 2.61614549789395435360235983313, 3.33185571109910878864208535461, 4.26833957638228030276239290218, 4.60597797100542210688415800585, 5.12025243774361693168193973260, 5.62334711197677024665142943185, 6.26975174528809199200487816086, 6.74623413736440975242841230138, 6.89767908990416466440617658362, 7.37665998210839182118262725385, 8.032007711018015513270642474185, 8.529988165970825784152675001891

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