↑

Properties

Label 4-728e2-1.1-c1e2-0-25
Degree $4$
Conductor $529984$
Sign $1$
Analytic cond. $33.7922$
Root an. cond. $2.41103$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s − 3·5-s + 7-s − 3·9-s − 2·11-s − 2·13-s + 6·15-s + 2·17-s − 6·19-s − 2·21-s + 5·25-s + 14·27-s + 9·29-s − 31-s + 4·33-s − 3·35-s − 10·37-s + 4·39-s − 3·41-s − 3·43-s + 9·45-s − 11·47-s − 6·49-s − 4·51-s − 3·53-s + 6·55-s + 12·57-s + ⋯
L(s)  = 1  − 1.15·3-s − 1.34·5-s + 0.377·7-s − 9-s − 0.603·11-s − 0.554·13-s + 1.54·15-s + 0.485·17-s − 1.37·19-s − 0.436·21-s + 25-s + 2.69·27-s + 1.67·29-s − 0.179·31-s + 0.696·33-s − 0.507·35-s − 1.64·37-s + 0.640·39-s − 0.468·41-s − 0.457·43-s + 1.34·45-s − 1.60·47-s − 6/7·49-s − 0.560·51-s − 0.412·53-s + 0.809·55-s + 1.58·57-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 529984 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 529984 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(529984\)    =    \(2^{6} \cdot 7^{2} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(33.7922\)
Root analytic conductor: \(2.41103\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 529984,\ (\ :1/2, 1/2),\ 1)\)

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 \( 1 \)
7$C_2$ \( 1 - T + p T^{2} \)
13$C_2$ \( 1 + 2 T + p T^{2} \)
good3$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.3.c_h
5$C_2^2$ \( 1 + 3 T + 4 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.5.d_e
11$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.11.c_x
17$C_2^2$ \( 1 - 2 T - 13 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.17.ac_an
19$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.19.g_bv
23$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.23.a_ax
29$C_2^2$ \( 1 - 9 T + 52 T^{2} - 9 p T^{3} + p^{2} T^{4} \) 2.29.aj_ca
31$C_2^2$ \( 1 + T - 30 T^{2} + p T^{3} + p^{2} T^{4} \) 2.31.b_abe
37$C_2$ \( ( 1 - T + p T^{2} )( 1 + 11 T + p T^{2} ) \) 2.37.k_cl
41$C_2^2$ \( 1 + 3 T - 32 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.41.d_abg
43$C_2^2$ \( 1 + 3 T - 34 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.43.d_abi
47$C_2^2$ \( 1 + 11 T + 74 T^{2} + 11 p T^{3} + p^{2} T^{4} \) 2.47.l_cw
53$C_2^2$ \( 1 + 3 T - 44 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.53.d_abs
59$C_2^2$ \( 1 + 12 T + 85 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.59.m_dh
61$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \) 2.61.k_fr
67$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \) 2.67.s_ih
71$C_2^2$ \( 1 - T - 70 T^{2} - p T^{3} + p^{2} T^{4} \) 2.71.ab_acs
73$C_2^2$ \( 1 + 11 T + 48 T^{2} + 11 p T^{3} + p^{2} T^{4} \) 2.73.l_bw
79$C_2^2$ \( 1 - 9 T + 2 T^{2} - 9 p T^{3} + p^{2} T^{4} \) 2.79.aj_c
83$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.83.q_iw
89$C_2^2$ \( 1 - 10 T + 11 T^{2} - 10 p T^{3} + p^{2} T^{4} \) 2.89.ak_l
97$C_2^2$ \( 1 - 13 T + 72 T^{2} - 13 p T^{3} + p^{2} T^{4} \) 2.97.an_cu
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

−10.43314165639347719582902835866, −10.00542970929772758021743164874, −9.212346144177994265292567663126, −8.699245434152593349296847277192, −8.362682670475705107528823623482, −8.182539007613774978172420930244, −7.54297387751119883840500787058, −7.20874132291086813603661315875, −6.36867670511174030911188506353, −6.34680427516949457631538116206, −5.74122411813187522605003405952, −5.01161062926751993932665747937, −4.71576316896358109364951279152, −4.60077667820527999883254340837, −3.40608782404698359156344392919, −3.21903736313216218490155977672, −2.50341705440163373610399575474, −1.45242600029716425330608173542, 0, 0, 1.45242600029716425330608173542, 2.50341705440163373610399575474, 3.21903736313216218490155977672, 3.40608782404698359156344392919, 4.60077667820527999883254340837, 4.71576316896358109364951279152, 5.01161062926751993932665747937, 5.74122411813187522605003405952, 6.34680427516949457631538116206, 6.36867670511174030911188506353, 7.20874132291086813603661315875, 7.54297387751119883840500787058, 8.182539007613774978172420930244, 8.362682670475705107528823623482, 8.699245434152593349296847277192, 9.212346144177994265292567663126, 10.00542970929772758021743164874, 10.43314165639347719582902835866

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