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Properties

Label 4-1568e2-1.1-c1e2-0-20
Degree $4$
Conductor $2458624$
Sign $1$
Analytic cond. $156.763$
Root an. cond. $3.53843$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s − 2·5-s − 9-s + 2·11-s − 4·15-s + 6·17-s + 10·19-s + 2·23-s + 25-s − 6·27-s + 14·31-s + 4·33-s + 6·37-s + 8·41-s − 8·43-s + 2·45-s + 18·47-s + 12·51-s − 2·53-s − 4·55-s + 20·57-s + 2·59-s − 6·61-s + 14·67-s + 4·69-s − 16·71-s + 18·73-s + ⋯
L(s)  = 1  + 1.15·3-s − 0.894·5-s − 1/3·9-s + 0.603·11-s − 1.03·15-s + 1.45·17-s + 2.29·19-s + 0.417·23-s + 1/5·25-s − 1.15·27-s + 2.51·31-s + 0.696·33-s + 0.986·37-s + 1.24·41-s − 1.21·43-s + 0.298·45-s + 2.62·47-s + 1.68·51-s − 0.274·53-s − 0.539·55-s + 2.64·57-s + 0.260·59-s − 0.768·61-s + 1.71·67-s + 0.481·69-s − 1.89·71-s + 2.10·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2458624 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2458624 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2458624\)    =    \(2^{10} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(156.763\)
Root analytic conductor: \(3.53843\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 2458624,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.852764170\)
\(L(\frac12)\) \(\approx\) \(3.852764170\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good3$D_{4}$ \( 1 - 2 T + 5 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.3.ac_f
5$D_{4}$ \( 1 + 2 T + 3 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.5.c_d
11$D_{4}$ \( 1 - 2 T + 21 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.11.ac_v
13$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \) 2.13.a_s
17$D_{4}$ \( 1 - 6 T + 35 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.17.ag_bj
19$D_{4}$ \( 1 - 10 T + 61 T^{2} - 10 p T^{3} + p^{2} T^{4} \) 2.19.ak_cj
23$D_{4}$ \( 1 - 2 T + 29 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.23.ac_bd
29$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \) 2.29.a_by
31$D_{4}$ \( 1 - 14 T + 109 T^{2} - 14 p T^{3} + p^{2} T^{4} \) 2.31.ao_ef
37$D_{4}$ \( 1 - 6 T + 51 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.37.ag_bz
41$D_{4}$ \( 1 - 8 T + 90 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.41.ai_dm
43$D_{4}$ \( 1 + 8 T + 70 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.43.i_cs
47$D_{4}$ \( 1 - 18 T + 173 T^{2} - 18 p T^{3} + p^{2} T^{4} \) 2.47.as_gr
53$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.53.c_ed
59$D_{4}$ \( 1 - 2 T + 21 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.59.ac_v
61$D_{4}$ \( 1 + 6 T + 99 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.61.g_dv
67$D_{4}$ \( 1 - 14 T + 165 T^{2} - 14 p T^{3} + p^{2} T^{4} \) 2.67.ao_gj
71$D_{4}$ \( 1 + 16 T + 174 T^{2} + 16 p T^{3} + p^{2} T^{4} \) 2.71.q_gs
73$D_{4}$ \( 1 - 18 T + 195 T^{2} - 18 p T^{3} + p^{2} T^{4} \) 2.73.as_hn
79$D_{4}$ \( 1 - 2 T + 109 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.79.ac_ef
83$D_{4}$ \( 1 - 8 T + 54 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.83.ai_cc
89$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \) 2.89.as_jz
97$D_{4}$ \( 1 - 8 T + 202 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.97.ai_hu
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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

−9.491144051014227641421894937708, −9.201722112760319807680275520998, −8.823801955153168895656416136666, −8.426941603886646841230903057950, −7.85046426055256123653913613611, −7.78174163656873312642551157027, −7.57108896925537878763901356580, −6.96877391641899939804844047838, −6.38521061736004752205058567798, −6.02565676268611011052871224095, −5.46970714943462433861773733396, −5.07369287099320617855733950743, −4.56995359275773683854200454313, −3.94214581971668857832698268013, −3.50672167662009372125494447989, −3.25164076735164495213499023057, −2.68183323877728961001328639015, −2.37952981554633394661957618764, −1.01951597661611130937101806782, −0.981611028434021842742737928419, 0.981611028434021842742737928419, 1.01951597661611130937101806782, 2.37952981554633394661957618764, 2.68183323877728961001328639015, 3.25164076735164495213499023057, 3.50672167662009372125494447989, 3.94214581971668857832698268013, 4.56995359275773683854200454313, 5.07369287099320617855733950743, 5.46970714943462433861773733396, 6.02565676268611011052871224095, 6.38521061736004752205058567798, 6.96877391641899939804844047838, 7.57108896925537878763901356580, 7.78174163656873312642551157027, 7.85046426055256123653913613611, 8.426941603886646841230903057950, 8.823801955153168895656416136666, 9.201722112760319807680275520998, 9.491144051014227641421894937708

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