Properties

Label 4-60e4-1.1-c1e2-0-16
Degree $4$
Conductor $12960000$
Sign $1$
Analytic cond. $826.340$
Root an. cond. $5.36154$
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  + 12·11-s − 8·19-s − 12·29-s + 8·31-s + 10·49-s − 12·59-s + 4·61-s − 24·71-s − 8·79-s + 24·89-s + 12·101-s − 4·109-s + 86·121-s + ⋯
L(s)  = 1  + 3.61·11-s − 1.83·19-s − 2.22·29-s + 1.43·31-s + 10/7·49-s − 1.56·59-s + 0.512·61-s − 2.84·71-s − 0.900·79-s + 2.54·89-s + 1.19·101-s − 0.383·109-s + 7.81·121-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 12960000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12960000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(12960000\)    =    \(2^{8} \cdot 3^{4} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(826.340\)
Root analytic conductor: \(5.36154\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 12960000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.153917922\)
\(L(\frac12)\) \(\approx\) \(3.153917922\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 \)
good7$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.7.a_ak
11$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.11.am_cg
13$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.13.a_ak
17$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.17.a_c
19$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.19.i_cc
23$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.23.a_abu
29$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.29.m_dq
31$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.31.ai_da
37$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.37.a_ak
41$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.41.a_de
43$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \) 2.43.a_aw
47$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.47.a_adq
53$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \) 2.53.a_acs
59$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.59.m_fy
61$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.61.ae_ew
67$C_2^2$ \( 1 - 118 T^{2} + p^{2} T^{4} \) 2.67.a_aeo
71$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.71.y_la
73$C_2^2$ \( 1 - 46 T^{2} + p^{2} T^{4} \) 2.73.a_abu
79$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.79.i_gs
83$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \) 2.83.a_aw
89$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \) 2.89.ay_mk
97$C_2^2$ \( 1 - 190 T^{2} + p^{2} T^{4} \) 2.97.a_ahi
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

−8.738563574983699868229627773746, −8.631257763503958944764484055077, −8.007547169977220550418858076983, −7.60768448406809336106818724476, −7.02058099779801638852536475747, −6.97124908232947533375272093629, −6.48855197973730516882011188312, −6.09327190919315350626284342643, −5.97793061647230281331957697206, −5.54790154231317012997678937115, −4.57249754922739729183021976310, −4.51360690271049199913910134070, −4.15627070691548670605831293873, −3.74098067043490529809660739880, −3.41892952375887439061413641771, −2.82916427336460765669770946059, −1.92425968585657447094574700080, −1.86056042187846097712020850355, −1.23255323987713508187576117340, −0.56330541303370341959236423617, 0.56330541303370341959236423617, 1.23255323987713508187576117340, 1.86056042187846097712020850355, 1.92425968585657447094574700080, 2.82916427336460765669770946059, 3.41892952375887439061413641771, 3.74098067043490529809660739880, 4.15627070691548670605831293873, 4.51360690271049199913910134070, 4.57249754922739729183021976310, 5.54790154231317012997678937115, 5.97793061647230281331957697206, 6.09327190919315350626284342643, 6.48855197973730516882011188312, 6.97124908232947533375272093629, 7.02058099779801638852536475747, 7.60768448406809336106818724476, 8.007547169977220550418858076983, 8.631257763503958944764484055077, 8.738563574983699868229627773746

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