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Properties

Label 4-810e2-1.1-c1e2-0-7
Degree $4$
Conductor $656100$
Sign $1$
Analytic cond. $41.8335$
Root an. cond. $2.54320$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 5-s − 5·7-s − 8-s − 10-s + 13-s − 5·14-s − 16-s + 12·17-s + 10·19-s + 9·23-s + 26-s + 4·31-s + 12·34-s + 5·35-s − 20·37-s + 10·38-s + 40-s + 3·41-s − 8·43-s + 9·46-s + 3·47-s + 7·49-s + 6·53-s + 5·56-s − 9·59-s − 8·61-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.447·5-s − 1.88·7-s − 0.353·8-s − 0.316·10-s + 0.277·13-s − 1.33·14-s − 1/4·16-s + 2.91·17-s + 2.29·19-s + 1.87·23-s + 0.196·26-s + 0.718·31-s + 2.05·34-s + 0.845·35-s − 3.28·37-s + 1.62·38-s + 0.158·40-s + 0.468·41-s − 1.21·43-s + 1.32·46-s + 0.437·47-s + 49-s + 0.824·53-s + 0.668·56-s − 1.17·59-s − 1.02·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 656100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 656100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(656100\)    =    \(2^{2} \cdot 3^{8} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(41.8335\)
Root analytic conductor: \(2.54320\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 656100,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.150598960\)
\(L(\frac12)\) \(\approx\) \(2.150598960\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - T + T^{2} \)
3 \( 1 \)
5$C_2$ \( 1 + T + T^{2} \)
good7$C_2$ \( ( 1 + T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.7.f_s
11$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.11.a_al
13$C_2^2$ \( 1 - T - 12 T^{2} - p T^{3} + p^{2} T^{4} \) 2.13.ab_am
17$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.17.am_cs
19$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \) 2.19.ak_cl
23$C_2^2$ \( 1 - 9 T + 58 T^{2} - 9 p T^{3} + p^{2} T^{4} \) 2.23.aj_cg
29$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.29.a_abd
31$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.31.ae_ap
37$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.37.u_gs
41$C_2^2$ \( 1 - 3 T - 32 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.41.ad_abg
43$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) 2.43.i_v
47$C_2^2$ \( 1 - 3 T - 38 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.47.ad_abm
53$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.53.ag_el
59$C_2^2$ \( 1 + 9 T + 22 T^{2} + 9 p T^{3} + p^{2} T^{4} \) 2.59.j_w
61$C_2^2$ \( 1 + 8 T + 3 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.61.i_d
67$C_2^2$ \( 1 - 4 T - 51 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.67.ae_abz
71$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.71.am_gw
73$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.73.ae_fu
79$C_2^2$ \( 1 + 2 T - 75 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.79.c_acx
83$C_2^2$ \( 1 + 6 T - 47 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.83.g_abv
89$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.89.am_ig
97$C_2^2$ \( 1 - 16 T + 159 T^{2} - 16 p T^{3} + p^{2} T^{4} \) 2.97.aq_gd
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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

−10.26896873734229316867876402430, −10.15231796779640940438503282521, −9.447242157207503129977352239370, −9.442520617230543651217767071609, −8.938090970001674792722162750578, −8.233822591569755889817394804188, −7.85800831563873960318949183031, −7.32376371177709433899643211379, −6.93796455900367642072885586469, −6.65505959738691563032227775667, −5.93778330737804887176041015384, −5.49650247410770706401530286960, −5.21909337888292808648466648450, −4.80445179715887463983548498128, −3.69171933039722432363116485971, −3.49963906805767146231647618942, −3.11612351563815677224340657144, −2.96920626315677779602152317095, −1.43997639641480665305737111686, −0.71801367938518459388409621191, 0.71801367938518459388409621191, 1.43997639641480665305737111686, 2.96920626315677779602152317095, 3.11612351563815677224340657144, 3.49963906805767146231647618942, 3.69171933039722432363116485971, 4.80445179715887463983548498128, 5.21909337888292808648466648450, 5.49650247410770706401530286960, 5.93778330737804887176041015384, 6.65505959738691563032227775667, 6.93796455900367642072885586469, 7.32376371177709433899643211379, 7.85800831563873960318949183031, 8.233822591569755889817394804188, 8.938090970001674792722162750578, 9.442520617230543651217767071609, 9.447242157207503129977352239370, 10.15231796779640940438503282521, 10.26896873734229316867876402430

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