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Properties

Label 4-92e4-1.1-c1e2-0-14
Degree $4$
Conductor $71639296$
Sign $1$
Analytic cond. $4567.78$
Root an. cond. $8.22103$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·3-s + 6·9-s − 10·13-s − 7·25-s − 4·27-s − 6·29-s + 16·31-s − 40·39-s − 18·41-s + 12·47-s − 2·49-s + 12·59-s − 12·71-s − 22·73-s − 28·75-s − 37·81-s − 24·87-s + 64·93-s + 6·101-s − 60·117-s − 10·121-s − 72·123-s + 127-s + 131-s + 137-s + 139-s + 48·141-s + ⋯
L(s)  = 1  + 2.30·3-s + 2·9-s − 2.77·13-s − 7/5·25-s − 0.769·27-s − 1.11·29-s + 2.87·31-s − 6.40·39-s − 2.81·41-s + 1.75·47-s − 2/7·49-s + 1.56·59-s − 1.42·71-s − 2.57·73-s − 3.23·75-s − 4.11·81-s − 2.57·87-s + 6.63·93-s + 0.597·101-s − 5.54·117-s − 0.909·121-s − 6.49·123-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 4.04·141-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 71639296 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 71639296 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(71639296\)    =    \(2^{8} \cdot 23^{4}\)
Sign: $1$
Analytic conductor: \(4567.78\)
Root analytic conductor: \(8.22103\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 71639296,\ (\ :1/2, 1/2),\ 1)\)

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 \( 1 \)
23 \( 1 \)
good3$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.3.ae_k
5$C_2^2$ \( 1 + 7 T^{2} + p^{2} T^{4} \) 2.5.a_h
7$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.7.a_c
11$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \) 2.11.a_k
13$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \) 2.13.k_bz
17$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \) 2.17.a_ao
19$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \) 2.19.a_ba
29$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.29.g_cp
31$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.31.aq_ew
37$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.37.a_cw
41$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \) 2.41.s_gh
43$C_2^2$ \( 1 + 38 T^{2} + p^{2} T^{4} \) 2.43.a_bm
47$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.47.am_fa
53$C_2^2$ \( 1 + 103 T^{2} + p^{2} T^{4} \) 2.53.a_dz
59$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.59.am_fy
61$C_2^2$ \( 1 + 95 T^{2} + p^{2} T^{4} \) 2.61.a_dr
67$C_2^2$ \( 1 + 86 T^{2} + p^{2} T^{4} \) 2.67.a_di
71$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.71.m_gw
73$C_2$ \( ( 1 + 11 T + p T^{2} )^{2} \) 2.73.w_kh
79$C_2^2$ \( 1 + 110 T^{2} + p^{2} T^{4} \) 2.79.a_eg
83$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.83.a_gk
89$C_2^2$ \( 1 + 175 T^{2} + p^{2} T^{4} \) 2.89.a_gt
97$C_2^2$ \( 1 + 47 T^{2} + p^{2} T^{4} \) 2.97.a_bv
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.62895164715173360183761853026, −7.46115700578524921125320039173, −7.08002787072192152029316228851, −6.93414360914174670556141217730, −6.11595479675485016706460193796, −6.06760826370947008502471329901, −5.41851873655352204898423681448, −5.08141442666137673795662028681, −4.79347122975735307494928663586, −4.31341891853917617677343012237, −3.89131910924881802239585761496, −3.66902944168378441381027200883, −3.11893261045253479996271717993, −2.66949487106070280373090640400, −2.57353632607832683726140659422, −2.28551586973661676352614618645, −1.77380623000668477939532424909, −1.27225741557779887381246428273, 0, 0, 1.27225741557779887381246428273, 1.77380623000668477939532424909, 2.28551586973661676352614618645, 2.57353632607832683726140659422, 2.66949487106070280373090640400, 3.11893261045253479996271717993, 3.66902944168378441381027200883, 3.89131910924881802239585761496, 4.31341891853917617677343012237, 4.79347122975735307494928663586, 5.08141442666137673795662028681, 5.41851873655352204898423681448, 6.06760826370947008502471329901, 6.11595479675485016706460193796, 6.93414360914174670556141217730, 7.08002787072192152029316228851, 7.46115700578524921125320039173, 7.62895164715173360183761853026

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