↑

Properties

Label 4-92e4-1.1-c1e2-0-10
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 $0$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 6·5-s + 4·7-s + 2·13-s + 12·19-s + 17·25-s + 6·29-s + 12·31-s + 24·35-s − 16·37-s + 6·41-s + 4·43-s − 8·47-s + 4·49-s + 2·53-s + 4·59-s − 2·61-s + 12·65-s + 8·67-s + 4·71-s + 6·73-s + 24·79-s − 9·81-s + 4·83-s + 2·89-s + 8·91-s + 72·95-s − 6·97-s + ⋯
L(s)  = 1  + 2.68·5-s + 1.51·7-s + 0.554·13-s + 2.75·19-s + 17/5·25-s + 1.11·29-s + 2.15·31-s + 4.05·35-s − 2.63·37-s + 0.937·41-s + 0.609·43-s − 1.16·47-s + 4/7·49-s + 0.274·53-s + 0.520·59-s − 0.256·61-s + 1.48·65-s + 0.977·67-s + 0.474·71-s + 0.702·73-s + 2.70·79-s − 81-s + 0.439·83-s + 0.211·89-s + 0.838·91-s + 7.38·95-s − 0.609·97-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: \(0\)
Selberg data: \((4,\ 71639296,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(11.71163337\)
\(L(\frac12)\) \(\approx\) \(11.71163337\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 + p^{2} T^{4} \) 2.3.a_a
5$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.5.ag_t
7$D_{4}$ \( 1 - 4 T + 12 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.7.ae_m
11$C_2^2$ \( 1 + 16 T^{2} + p^{2} T^{4} \) 2.11.a_q
13$D_{4}$ \( 1 - 2 T + 3 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.13.ac_d
17$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \) 2.17.a_k
19$D_{4}$ \( 1 - 12 T + 68 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.19.am_cq
29$D_{4}$ \( 1 - 6 T + 43 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.29.ag_br
31$D_{4}$ \( 1 - 12 T + 74 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.31.am_cw
37$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.37.q_fi
41$D_{4}$ \( 1 - 6 T + 67 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.41.ag_cp
43$D_{4}$ \( 1 - 4 T + 66 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.43.ae_co
47$D_{4}$ \( 1 + 8 T + 104 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.47.i_ea
53$D_{4}$ \( 1 - 2 T + 11 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.53.ac_l
59$D_{4}$ \( 1 - 4 T + 68 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.59.ae_cq
61$D_{4}$ \( 1 + 2 T + 99 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.61.c_dv
67$D_{4}$ \( 1 - 8 T + 54 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.67.ai_cc
71$D_{4}$ \( 1 - 4 T + 140 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.71.ae_fk
73$D_{4}$ \( 1 - 6 T + 131 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.73.ag_fb
79$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \) 2.79.ay_lq
83$D_{4}$ \( 1 - 4 T + 146 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.83.ae_fq
89$D_{4}$ \( 1 - 2 T + 83 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.89.ac_df
97$D_{4}$ \( 1 + 6 T + 179 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.97.g_gx
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

−7.86668706319201402607435040472, −7.80607509953733529604225945752, −7.23836799880599484556609566313, −6.63906161976185981370863286895, −6.62950779471346565651457428053, −6.30156209584082815055735961735, −5.77228564325791053251520308822, −5.37479390080058446999581110965, −5.23456352324666812942784345385, −5.16056214481496802287876337178, −4.61925180921018203460163139079, −4.23336706997867992943812301132, −3.41111370659399098975595848288, −3.34749610682031238008059064562, −2.64843514858767870235544348608, −2.35699113310480059214780586893, −1.98927943517059896209008864544, −1.37831102758281447525282821381, −1.25478938545719888037981672269, −0.850958519017503485198336220236, 0.850958519017503485198336220236, 1.25478938545719888037981672269, 1.37831102758281447525282821381, 1.98927943517059896209008864544, 2.35699113310480059214780586893, 2.64843514858767870235544348608, 3.34749610682031238008059064562, 3.41111370659399098975595848288, 4.23336706997867992943812301132, 4.61925180921018203460163139079, 5.16056214481496802287876337178, 5.23456352324666812942784345385, 5.37479390080058446999581110965, 5.77228564325791053251520308822, 6.30156209584082815055735961735, 6.62950779471346565651457428053, 6.63906161976185981370863286895, 7.23836799880599484556609566313, 7.80607509953733529604225945752, 7.86668706319201402607435040472

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