↑

Properties

Label 4-208e2-1.1-c1e2-0-7
Degree $4$
Conductor $43264$
Sign $1$
Analytic cond. $2.75855$
Root an. cond. $1.28875$
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  − 2·5-s + 4·7-s + 3·9-s + 4·11-s + 7·13-s − 3·17-s − 4·23-s − 7·25-s + 29-s − 8·31-s − 8·35-s − 3·37-s + 9·41-s − 8·43-s − 6·45-s + 16·47-s + 7·49-s − 18·53-s − 8·55-s − 4·59-s − 7·61-s + 12·63-s − 14·65-s + 4·67-s − 8·71-s + 22·73-s + 16·77-s + ⋯
L(s)  = 1  − 0.894·5-s + 1.51·7-s + 9-s + 1.20·11-s + 1.94·13-s − 0.727·17-s − 0.834·23-s − 7/5·25-s + 0.185·29-s − 1.43·31-s − 1.35·35-s − 0.493·37-s + 1.40·41-s − 1.21·43-s − 0.894·45-s + 2.33·47-s + 49-s − 2.47·53-s − 1.07·55-s − 0.520·59-s − 0.896·61-s + 1.51·63-s − 1.73·65-s + 0.488·67-s − 0.949·71-s + 2.57·73-s + 1.82·77-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43264 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43264 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(43264\)    =    \(2^{8} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(2.75855\)
Root analytic conductor: \(1.28875\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 43264,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.574943349\)
\(L(\frac12)\) \(\approx\) \(1.574943349\)
\(L(\frac{3}{2})\) 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 \( 1 \)
13$C_2$ \( 1 - 7 T + p T^{2} \)
good3$C_2$ \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \) 2.3.a_ad
5$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.5.c_l
7$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + T + p T^{2} ) \) 2.7.ae_j
11$C_2^2$ \( 1 - 4 T + 5 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.11.ae_f
17$C_2^2$ \( 1 + 3 T - 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.17.d_ai
19$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.19.a_at
23$C_2^2$ \( 1 + 4 T - 7 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.23.e_ah
29$C_2^2$ \( 1 - T - 28 T^{2} - p T^{3} + p^{2} T^{4} \) 2.29.ab_abc
31$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.31.i_da
37$C_2^2$ \( 1 + 3 T - 28 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.37.d_abc
41$C_2^2$ \( 1 - 9 T + 40 T^{2} - 9 p T^{3} + p^{2} T^{4} \) 2.41.aj_bo
43$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) 2.43.i_v
47$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.47.aq_gc
53$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \) 2.53.s_hf
59$C_2^2$ \( 1 + 4 T - 43 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.59.e_abr
61$C_2^2$ \( 1 + 7 T - 12 T^{2} + 7 p T^{3} + p^{2} T^{4} \) 2.61.h_am
67$C_2^2$ \( 1 - 4 T - 51 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.67.ae_abz
71$C_2^2$ \( 1 + 8 T - 7 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.71.i_ah
73$C_2$ \( ( 1 - 11 T + p T^{2} )^{2} \) 2.73.aw_kh
79$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.79.ai_gs
83$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.83.a_gk
89$C_2^2$ \( 1 - 6 T - 53 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.89.ag_acb
97$C_2^2$ \( 1 + 2 T - 93 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.97.c_adp
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

−12.31254166152761015895881843902, −12.25196933517873443161541911548, −11.46712237280786853388564830075, −11.27538744150582375657188587471, −10.85714630926024950776075283777, −10.45902203114330446642378420660, −9.425870767464324672988072167903, −9.278124938169969448763987916790, −8.512862190502260882779457285438, −8.163339103320999247101422951462, −7.61653422691277270803106023594, −7.26066693892898955863983143934, −6.29156876859747315138332365657, −6.12242102580676883025405456319, −5.12611426549667021393662749119, −4.40410805845979791476760715651, −3.85528860880685358985131670949, −3.71877137429415820204304026776, −1.95416902815962143060931653700, −1.37657063688606665036298754407, 1.37657063688606665036298754407, 1.95416902815962143060931653700, 3.71877137429415820204304026776, 3.85528860880685358985131670949, 4.40410805845979791476760715651, 5.12611426549667021393662749119, 6.12242102580676883025405456319, 6.29156876859747315138332365657, 7.26066693892898955863983143934, 7.61653422691277270803106023594, 8.163339103320999247101422951462, 8.512862190502260882779457285438, 9.278124938169969448763987916790, 9.425870767464324672988072167903, 10.45902203114330446642378420660, 10.85714630926024950776075283777, 11.27538744150582375657188587471, 11.46712237280786853388564830075, 12.25196933517873443161541911548, 12.31254166152761015895881843902

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