↑

Properties

Label 4-92e4-1.1-c1e2-0-7
Degree $4$
Conductor $71639296$
Sign $1$
Analytic cond. $4567.78$
Root an. cond. $8.22103$
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 − 6·7-s − 6·11-s + 2·13-s + 6·19-s + 12·21-s − 7·25-s + 2·27-s + 6·29-s + 4·31-s + 12·33-s + 12·37-s − 4·39-s − 6·41-s + 12·43-s − 18·47-s + 16·49-s − 12·53-s − 12·57-s − 6·59-s + 12·61-s + 12·67-s − 18·71-s − 10·73-s + 14·75-s + 36·77-s + 12·79-s + ⋯
L(s)  = 1  − 1.15·3-s − 2.26·7-s − 1.80·11-s + 0.554·13-s + 1.37·19-s + 2.61·21-s − 7/5·25-s + 0.384·27-s + 1.11·29-s + 0.718·31-s + 2.08·33-s + 1.97·37-s − 0.640·39-s − 0.937·41-s + 1.82·43-s − 2.62·47-s + 16/7·49-s − 1.64·53-s − 1.58·57-s − 0.781·59-s + 1.53·61-s + 1.46·67-s − 2.13·71-s − 1.17·73-s + 1.61·75-s + 4.10·77-s + 1.35·79-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 71639296 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 71639296 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(71639296\)    =    \(2^{8} \cdot 23^{4}\)
Sign: $1$
Analytic conductor: \(4567.78\)
Root analytic conductor: \(8.22103\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 71639296,\ (\ :1/2, 1/2),\ 1)\)

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 \( 1 \)
23 \( 1 \)
good3$D_{4}$ \( 1 + 2 T + 4 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.3.c_e
5$C_2^2$ \( 1 + 7 T^{2} + p^{2} T^{4} \) 2.5.a_h
7$D_{4}$ \( 1 + 6 T + 20 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.7.g_u
11$D_{4}$ \( 1 + 6 T + 28 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.11.g_bc
13$C_2$ \( ( 1 - T + p T^{2} )^{2} \) 2.13.ac_bb
17$C_2^2$ \( 1 + 22 T^{2} + p^{2} T^{4} \) 2.17.a_w
19$D_{4}$ \( 1 - 6 T + 44 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.19.ag_bs
29$D_{4}$ \( 1 - 6 T + 55 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.29.ag_cd
31$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.31.ae_co
37$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.37.am_eg
41$D_{4}$ \( 1 + 6 T + 79 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.41.g_db
43$D_{4}$ \( 1 - 12 T + 74 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.43.am_cw
47$D_{4}$ \( 1 + 18 T + 172 T^{2} + 18 p T^{3} + p^{2} T^{4} \) 2.47.s_gq
53$D_{4}$ \( 1 + 12 T + 139 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.53.m_fj
59$D_{4}$ \( 1 + 6 T + 100 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.59.g_dw
61$D_{4}$ \( 1 - 12 T + 131 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.61.am_fb
67$D_{4}$ \( 1 - 12 T + 158 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.67.am_gc
71$D_{4}$ \( 1 + 18 T + 220 T^{2} + 18 p T^{3} + p^{2} T^{4} \) 2.71.s_im
73$D_{4}$ \( 1 + 10 T + 63 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.73.k_cl
79$D_{4}$ \( 1 - 12 T + 182 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.79.am_ha
83$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.83.m_hu
89$D_{4}$ \( 1 - 12 T + 211 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.89.am_id
97$C_2^2$ \( 1 + 191 T^{2} + p^{2} T^{4} \) 2.97.a_hj
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

−7.49198527214998320688940296303, −7.41538642337793150307342865092, −6.60223633672498563201326819062, −6.55549958810734222654500844059, −6.13959442611324377533971448688, −5.99148617154458100149320207462, −5.68103668426306045234361699205, −5.31087706078447029514408473977, −4.82175953668445339920074222226, −4.67637557874709123987660425647, −4.07310481050417910323323515527, −3.50325238410681928375601199212, −3.18245809036786745834611815914, −3.05416913114154316068823557074, −2.53044758228176117503294949430, −2.16055259226067040659914028898, −1.24340201134382367266211849412, −0.75973696621661718038313681159, 0, 0, 0.75973696621661718038313681159, 1.24340201134382367266211849412, 2.16055259226067040659914028898, 2.53044758228176117503294949430, 3.05416913114154316068823557074, 3.18245809036786745834611815914, 3.50325238410681928375601199212, 4.07310481050417910323323515527, 4.67637557874709123987660425647, 4.82175953668445339920074222226, 5.31087706078447029514408473977, 5.68103668426306045234361699205, 5.99148617154458100149320207462, 6.13959442611324377533971448688, 6.55549958810734222654500844059, 6.60223633672498563201326819062, 7.41538642337793150307342865092, 7.49198527214998320688940296303

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