↑

Properties

Label 4-72e4-1.1-c1e2-0-23
Degree $4$
Conductor $26873856$
Sign $1$
Analytic cond. $1713.50$
Root an. cond. $6.43385$
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  + 4·5-s − 6·13-s − 8·17-s + 5·25-s − 4·29-s − 2·37-s − 16·41-s − 14·49-s + 8·53-s − 10·61-s − 24·65-s − 6·73-s − 32·85-s − 16·89-s − 36·97-s + 40·101-s − 6·109-s − 16·113-s − 22·121-s − 4·125-s + 127-s + 131-s + 137-s + 139-s − 16·145-s + 149-s + 151-s + ⋯
L(s)  = 1  + 1.78·5-s − 1.66·13-s − 1.94·17-s + 25-s − 0.742·29-s − 0.328·37-s − 2.49·41-s − 2·49-s + 1.09·53-s − 1.28·61-s − 2.97·65-s − 0.702·73-s − 3.47·85-s − 1.69·89-s − 3.65·97-s + 3.98·101-s − 0.574·109-s − 1.50·113-s − 2·121-s − 0.357·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 1.32·145-s + 0.0819·149-s + 0.0813·151-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 26873856 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26873856 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(26873856\)    =    \(2^{12} \cdot 3^{8}\)
Sign: $1$
Analytic conductor: \(1713.50\)
Root analytic conductor: \(6.43385\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 26873856,\ (\ :1/2, 1/2),\ 1)\)

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 \( 1 \)
good5$C_2^2$ \( 1 - 4 T + 11 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.5.ae_l
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$C_2^2$ \( 1 + 6 T + 23 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.13.g_x
17$C_2^2$ \( 1 + 8 T + 47 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.17.i_bv
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$C_2^2$ \( 1 + 4 T - 13 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.29.e_an
31$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.31.a_ck
37$C_2^2$ \( 1 + 2 T - 33 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.37.c_abh
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$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.53.ai_es
59$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.59.a_eo
61$C_2^2$ \( 1 + 10 T + 39 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.61.k_bn
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$C_2^2$ \( 1 + 6 T - 37 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.73.g_abl
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$C_2^2$ \( 1 + 16 T + 167 T^{2} + 16 p T^{3} + p^{2} T^{4} \) 2.89.q_gl
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

−8.023365539524503817272813724664, −7.71267446315760675239263482367, −7.08186836506039024620719054564, −6.92961408862276736946718376976, −6.60235172455684565467584039728, −6.34803488014065279847960095774, −5.70731805204780279287059727512, −5.61128448007755988453591823625, −4.98618410760750856941749296442, −4.98476160074597816712080145381, −4.36479963795145446667101758256, −4.09859903820601502089183799265, −3.32250932785269187031862956978, −3.00171302715781682430822090693, −2.34082153257008314088890328191, −2.24997742222037005069043774740, −1.68378626050903874069642452345, −1.42599352714473386334126094148, 0, 0, 1.42599352714473386334126094148, 1.68378626050903874069642452345, 2.24997742222037005069043774740, 2.34082153257008314088890328191, 3.00171302715781682430822090693, 3.32250932785269187031862956978, 4.09859903820601502089183799265, 4.36479963795145446667101758256, 4.98476160074597816712080145381, 4.98618410760750856941749296442, 5.61128448007755988453591823625, 5.70731805204780279287059727512, 6.34803488014065279847960095774, 6.60235172455684565467584039728, 6.92961408862276736946718376976, 7.08186836506039024620719054564, 7.71267446315760675239263482367, 8.023365539524503817272813724664

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