↑

Properties

Label 4-8820e2-1.1-c1e2-0-8
Degree $4$
Conductor $77792400$
Sign $1$
Analytic cond. $4960.11$
Root an. cond. $8.39214$
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  − 2·5-s + 4·11-s − 8·17-s − 8·19-s + 12·23-s + 3·25-s + 4·29-s − 4·37-s − 4·41-s − 8·43-s − 4·47-s + 12·53-s − 8·55-s − 8·67-s + 4·71-s + 8·73-s − 20·79-s − 4·83-s + 16·85-s − 4·89-s + 16·95-s − 8·97-s + 12·101-s + 4·107-s − 20·109-s + 4·113-s − 24·115-s + ⋯
L(s)  = 1  − 0.894·5-s + 1.20·11-s − 1.94·17-s − 1.83·19-s + 2.50·23-s + 3/5·25-s + 0.742·29-s − 0.657·37-s − 0.624·41-s − 1.21·43-s − 0.583·47-s + 1.64·53-s − 1.07·55-s − 0.977·67-s + 0.474·71-s + 0.936·73-s − 2.25·79-s − 0.439·83-s + 1.73·85-s − 0.423·89-s + 1.64·95-s − 0.812·97-s + 1.19·101-s + 0.386·107-s − 1.91·109-s + 0.376·113-s − 2.23·115-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 77792400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 77792400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(77792400\)    =    \(2^{4} \cdot 3^{4} \cdot 5^{2} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(4960.11\)
Root analytic conductor: \(8.39214\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 77792400,\ (\ :1/2, 1/2),\ 1)\)

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 \( 1 \)
3 \( 1 \)
5$C_1$ \( ( 1 + T )^{2} \)
7 \( 1 \)
good11$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.11.ae_ba
13$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \) 2.13.a_s
17$D_{4}$ \( 1 + 8 T + 42 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.17.i_bq
19$D_{4}$ \( 1 + 8 T + 52 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.19.i_ca
23$D_{4}$ \( 1 - 12 T + 80 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.23.am_dc
29$D_{4}$ \( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.29.ae_be
31$C_2^2$ \( 1 + 44 T^{2} + p^{2} T^{4} \) 2.31.a_bs
37$D_{4}$ \( 1 + 4 T + 6 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.37.e_g
41$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.41.e_di
43$D_{4}$ \( 1 + 8 T + 70 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.43.i_cs
47$D_{4}$ \( 1 + 4 T + 26 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.47.e_ba
53$D_{4}$ \( 1 - 12 T + 140 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.53.am_fk
59$C_2^2$ \( 1 + 110 T^{2} + p^{2} T^{4} \) 2.59.a_eg
61$C_2^2$ \( 1 + 24 T^{2} + p^{2} T^{4} \) 2.61.a_y
67$D_{4}$ \( 1 + 8 T + 142 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.67.i_fm
71$D_{4}$ \( 1 - 4 T + 74 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.71.ae_cw
73$D_{4}$ \( 1 - 8 T + 90 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.73.ai_dm
79$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.79.u_jy
83$D_{4}$ \( 1 + 4 T + 162 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.83.e_gg
89$D_{4}$ \( 1 + 4 T + 174 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.89.e_gs
97$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.97.i_ic
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

−7.34855475767127988090085173941, −7.13304885736852915700009476617, −6.93779523868016011400576679904, −6.63061868617340726001030699476, −6.35091391678578719311638554240, −6.09547868026743123145180948196, −5.31892449807896449207416508135, −5.14140241972054827041610258625, −4.55576621943235263292608948608, −4.54872442672678837514344424404, −4.04686916712369959486742807984, −3.78695530331955070755437266275, −3.26400504979946733250776653273, −2.94229646436818046960609102709, −2.34186841390727831409701225077, −2.12081502815938201861201496494, −1.25718657991151184932749391456, −1.16809601989841580868460910268, 0, 0, 1.16809601989841580868460910268, 1.25718657991151184932749391456, 2.12081502815938201861201496494, 2.34186841390727831409701225077, 2.94229646436818046960609102709, 3.26400504979946733250776653273, 3.78695530331955070755437266275, 4.04686916712369959486742807984, 4.54872442672678837514344424404, 4.55576621943235263292608948608, 5.14140241972054827041610258625, 5.31892449807896449207416508135, 6.09547868026743123145180948196, 6.35091391678578719311638554240, 6.63061868617340726001030699476, 6.93779523868016011400576679904, 7.13304885736852915700009476617, 7.34855475767127988090085173941

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