Properties

Label 4-442368-1.1-c1e2-0-26
Degree $4$
Conductor $442368$
Sign $1$
Analytic cond. $28.2057$
Root an. cond. $2.30454$
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  − 3-s + 9-s + 8·11-s + 4·13-s + 8·23-s + 6·25-s − 27-s − 8·33-s − 4·37-s − 4·39-s + 8·47-s − 10·49-s − 8·59-s + 28·61-s − 8·69-s + 24·71-s − 20·73-s − 6·75-s + 81-s − 24·83-s + 20·97-s + 8·99-s + 8·107-s − 12·109-s + 4·111-s + 4·117-s + 26·121-s + ⋯
L(s)  = 1  − 0.577·3-s + 1/3·9-s + 2.41·11-s + 1.10·13-s + 1.66·23-s + 6/5·25-s − 0.192·27-s − 1.39·33-s − 0.657·37-s − 0.640·39-s + 1.16·47-s − 1.42·49-s − 1.04·59-s + 3.58·61-s − 0.963·69-s + 2.84·71-s − 2.34·73-s − 0.692·75-s + 1/9·81-s − 2.63·83-s + 2.03·97-s + 0.804·99-s + 0.773·107-s − 1.14·109-s + 0.379·111-s + 0.369·117-s + 2.36·121-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 442368 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 442368 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(442368\)    =    \(2^{14} \cdot 3^{3}\)
Sign: $1$
Analytic conductor: \(28.2057\)
Root analytic conductor: \(2.30454\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 442368,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.214216501\)
\(L(\frac12)\) \(\approx\) \(2.214216501\)
\(L(\frac{3}{2})\) 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$C_1$ \( 1 + T \)
good5$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.5.a_ag
7$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.7.a_k
11$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.11.ai_bm
13$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.13.ae_be
17$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.17.a_be
19$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.19.a_aba
23$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.23.ai_ck
29$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.29.a_cg
31$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.31.a_ba
37$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.37.e_da
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.41.a_bu
43$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.43.a_di
47$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.47.ai_eg
53$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.53.a_ec
59$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.59.i_fe
61$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \) 2.61.abc_mg
67$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.67.a_eo
71$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \) 2.71.ay_la
73$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.73.u_jm
79$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.79.a_cg
83$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.83.y_ly
89$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.89.a_as
97$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.97.au_li
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

−8.551716675388505933005243425858, −8.472401352118263853462982524884, −7.50551074140987907827478405271, −6.89547356553480728554873620104, −6.84130771729387693507813779002, −6.39174188210860038385829426400, −5.83450410447989994046324048602, −5.35620126632427385957975038673, −4.69490327290429029526959386590, −4.30701207983705537043099547844, −3.49343359434470738584054327342, −3.47403249961345600185284818653, −2.35029669656051858459889649700, −1.28088417651653847704023663777, −1.06515660853035145988959375788, 1.06515660853035145988959375788, 1.28088417651653847704023663777, 2.35029669656051858459889649700, 3.47403249961345600185284818653, 3.49343359434470738584054327342, 4.30701207983705537043099547844, 4.69490327290429029526959386590, 5.35620126632427385957975038673, 5.83450410447989994046324048602, 6.39174188210860038385829426400, 6.84130771729387693507813779002, 6.89547356553480728554873620104, 7.50551074140987907827478405271, 8.472401352118263853462982524884, 8.551716675388505933005243425858

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