↑

Properties

Label 4-165888-1.1-c1e2-0-6
Degree $4$
Conductor $165888$
Sign $1$
Analytic cond. $10.5771$
Root an. cond. $1.80340$
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  + 16·17-s + 6·25-s − 16·41-s − 14·49-s + 12·73-s + 32·89-s − 36·97-s + 32·113-s − 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 10·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯
L(s)  = 1  + 3.88·17-s + 6/5·25-s − 2.49·41-s − 2·49-s + 1.40·73-s + 3.39·89-s − 3.65·97-s + 3.01·113-s − 2·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 0.769·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 165888 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 165888 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(165888\)    =    \(2^{11} \cdot 3^{4}\)
Sign: $1$
Analytic conductor: \(10.5771\)
Root analytic conductor: \(1.80340\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 165888,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.871188571\)
\(L(\frac12)\) \(\approx\) \(1.871188571\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good5$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.5.a_ag
7$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.7.a_o
11$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.11.a_w
13$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.13.a_ak
17$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.17.aq_du
19$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.19.a_bm
23$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.23.a_bu
29$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.29.a_bq
31$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.31.a_ck
37$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.37.a_cs
41$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.41.q_fq
43$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.43.a_di
47$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.47.a_dq
53$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.53.a_dm
59$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.59.a_eo
61$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.61.a_w
67$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.67.a_fe
71$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.71.a_fm
73$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.73.am_ha
79$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.79.a_gc
83$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.83.a_gk
89$C_2$ \( ( 1 - 16 T + p T^{2} )^{2} \) 2.89.abg_qs
97$C_2$ \( ( 1 + 18 T + p T^{2} )^{2} \) 2.97.bk_ty
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

−9.372732192062624224981148085357, −8.612772853079119873302234581371, −8.142751074316513789502919252186, −7.909987862286350183375095910259, −7.33244289349889698794346569795, −6.81411763071537323600840753231, −6.25541731904712913819044308910, −5.66990658847027277525585177583, −5.02525715711667990043884728260, −5.01237677697104957198001932438, −3.81076717999348126429094994331, −3.29164018328411348390302871759, −3.02473421793484999957631922902, −1.74502909344688296171382532979, −1.00089337770198902241003598585, 1.00089337770198902241003598585, 1.74502909344688296171382532979, 3.02473421793484999957631922902, 3.29164018328411348390302871759, 3.81076717999348126429094994331, 5.01237677697104957198001932438, 5.02525715711667990043884728260, 5.66990658847027277525585177583, 6.25541731904712913819044308910, 6.81411763071537323600840753231, 7.33244289349889698794346569795, 7.909987862286350183375095910259, 8.142751074316513789502919252186, 8.612772853079119873302234581371, 9.372732192062624224981148085357

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