↑

Properties

Label 4-266240-1.1-c1e2-0-22
Degree $4$
Conductor $266240$
Sign $-1$
Analytic cond. $16.9756$
Root an. cond. $2.02981$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 5-s + 2·9-s − 13-s − 10·17-s − 4·25-s + 8·37-s − 14·41-s + 2·45-s − 8·49-s − 16·53-s − 65-s − 5·81-s − 10·85-s + 4·89-s − 10·97-s + 8·101-s + 14·109-s + 6·113-s − 2·117-s − 2·121-s − 4·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯
L(s)  = 1  + 0.447·5-s + 2/3·9-s − 0.277·13-s − 2.42·17-s − 4/5·25-s + 1.31·37-s − 2.18·41-s + 0.298·45-s − 8/7·49-s − 2.19·53-s − 0.124·65-s − 5/9·81-s − 1.08·85-s + 0.423·89-s − 1.01·97-s + 0.796·101-s + 1.34·109-s + 0.564·113-s − 0.184·117-s − 0.181·121-s − 0.357·125-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 + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 266240 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 266240 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(266240\)    =    \(2^{12} \cdot 5 \cdot 13\)
Sign: $-1$
Analytic conductor: \(16.9756\)
Root analytic conductor: \(2.02981\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 266240,\ (\ :1/2, 1/2),\ -1)\)

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 \( 1 \)
5$C_1$$\times$$C_2$ \( ( 1 - T )( 1 + p T^{2} ) \)
13$C_1$$\times$$C_2$ \( ( 1 + T )( 1 + p T^{2} ) \)
good3$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.3.a_ac
7$C_2^2$ \( 1 + 8 T^{2} + p^{2} T^{4} \) 2.7.a_i
11$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.11.a_c
17$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.17.k_by
19$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.19.a_ak
23$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \) 2.23.a_o
29$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.29.a_cc
31$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.31.a_ba
37$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.37.ai_cc
41$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.41.o_ec
43$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \) 2.43.a_ba
47$C_2^2$ \( 1 + 32 T^{2} + p^{2} T^{4} \) 2.47.a_bg
53$C_2$$\times$$C_2$ \( ( 1 + 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.53.q_gk
59$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.59.a_ac
61$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.61.a_di
67$C_2^2$ \( 1 + 120 T^{2} + p^{2} T^{4} \) 2.67.a_eq
71$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.71.a_fi
73$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.73.a_aby
79$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \) 2.79.a_be
83$C_2^2$ \( 1 + 76 T^{2} + p^{2} T^{4} \) 2.83.a_cy
89$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.89.ae_bm
97$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.97.k_ic
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

−8.814948954310619944961127021769, −8.130080066216507765520363362211, −7.84948064895246259220743713533, −7.09748325913398788732485404507, −6.80650763665918847170656863806, −6.25210229302241560720611045318, −6.00397993466679958533029788463, −5.01674357789646734908828822930, −4.77405587998275819489172691150, −4.25340642747902156484584729630, −3.59817374031002401827035232395, −2.81732365589265577864519786969, −2.08654576833532343336773541478, −1.58624489229399821285878881237, 0, 1.58624489229399821285878881237, 2.08654576833532343336773541478, 2.81732365589265577864519786969, 3.59817374031002401827035232395, 4.25340642747902156484584729630, 4.77405587998275819489172691150, 5.01674357789646734908828822930, 6.00397993466679958533029788463, 6.25210229302241560720611045318, 6.80650763665918847170656863806, 7.09748325913398788732485404507, 7.84948064895246259220743713533, 8.130080066216507765520363362211, 8.814948954310619944961127021769

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