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Properties

Label 4-800e2-1.1-c1e2-0-47
Degree $4$
Conductor $640000$
Sign $1$
Analytic cond. $40.8069$
Root an. cond. $2.52745$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s − 2·7-s + 2·9-s − 8·13-s + 8·17-s − 8·19-s + 4·21-s − 10·23-s − 6·27-s + 16·39-s − 8·41-s − 14·43-s − 6·47-s + 2·49-s − 16·51-s − 8·53-s + 16·57-s − 8·59-s − 16·61-s − 4·63-s + 6·67-s + 20·69-s − 8·73-s + 16·79-s + 11·81-s + 10·83-s + 16·91-s + ⋯
L(s)  = 1  − 1.15·3-s − 0.755·7-s + 2/3·9-s − 2.21·13-s + 1.94·17-s − 1.83·19-s + 0.872·21-s − 2.08·23-s − 1.15·27-s + 2.56·39-s − 1.24·41-s − 2.13·43-s − 0.875·47-s + 2/7·49-s − 2.24·51-s − 1.09·53-s + 2.11·57-s − 1.04·59-s − 2.04·61-s − 0.503·63-s + 0.733·67-s + 2.40·69-s − 0.936·73-s + 1.80·79-s + 11/9·81-s + 1.09·83-s + 1.67·91-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 640000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 640000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(640000\)    =    \(2^{10} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(40.8069\)
Root analytic conductor: \(2.52745\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 640000,\ (\ :1/2, 1/2),\ 1)\)

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 \( 1 \)
5 \( 1 \)
good3$C_2^2$ \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.3.c_c
7$C_2^2$ \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.7.c_c
11$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \) 2.11.a_ag
13$C_2^2$ \( 1 + 8 T + 32 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.13.i_bg
17$C_2^2$ \( 1 - 8 T + 32 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.17.ai_bg
19$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.19.i_cc
23$C_2^2$ \( 1 + 10 T + 50 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.23.k_by
29$C_2^2$ \( 1 - 54 T^{2} + p^{2} T^{4} \) 2.29.a_acc
31$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.31.a_c
37$C_2^2$ \( 1 + p^{2} T^{4} \) 2.37.a_a
41$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.41.i_du
43$C_2^2$ \( 1 + 14 T + 98 T^{2} + 14 p T^{3} + p^{2} T^{4} \) 2.43.o_du
47$C_2^2$ \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.47.g_s
53$C_2^2$ \( 1 + 8 T + 32 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.53.i_bg
59$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.59.i_fe
61$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.61.q_he
67$C_2^2$ \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.67.ag_s
71$C_2^2$ \( 1 + 114 T^{2} + p^{2} T^{4} \) 2.71.a_ek
73$C_2^2$ \( 1 + 8 T + 32 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.73.i_bg
79$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.79.aq_io
83$C_2^2$ \( 1 - 10 T + 50 T^{2} - 10 p T^{3} + p^{2} T^{4} \) 2.83.ak_by
89$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) 2.89.a_ada
97$C_2^2$ \( 1 + 24 T + 288 T^{2} + 24 p T^{3} + p^{2} T^{4} \) 2.97.y_lc
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.919772862619133434876702545280, −9.823472163734552459492393067794, −9.574661109830573488886018331404, −8.836036840990811156891313534127, −8.124811921448576934330753426699, −7.88530630239642942368852942648, −7.53018700744890584918938106709, −6.89001267097818063520236380118, −6.32401712171205911247656397916, −6.30755155853662791263111575752, −5.58471461078796806943041311876, −5.22538638667272582894854542515, −4.70776300480667285909616321813, −4.28491194584830605190006286174, −3.48644310466134657736818162271, −3.11864059326382587524087302816, −2.13376328614016300035544002000, −1.69971321921887323016799424782, 0, 0, 1.69971321921887323016799424782, 2.13376328614016300035544002000, 3.11864059326382587524087302816, 3.48644310466134657736818162271, 4.28491194584830605190006286174, 4.70776300480667285909616321813, 5.22538638667272582894854542515, 5.58471461078796806943041311876, 6.30755155853662791263111575752, 6.32401712171205911247656397916, 6.89001267097818063520236380118, 7.53018700744890584918938106709, 7.88530630239642942368852942648, 8.124811921448576934330753426699, 8.836036840990811156891313534127, 9.574661109830573488886018331404, 9.823472163734552459492393067794, 9.919772862619133434876702545280

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