Properties

Label 4-648e2-1.1-c1e2-0-35
Degree $4$
Conductor $419904$
Sign $1$
Analytic cond. $26.7734$
Root an. cond. $2.27471$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + 2·4-s + 8·7-s + 16·14-s − 4·16-s + 10·17-s + 4·23-s + 6·25-s + 16·28-s − 8·31-s − 8·32-s + 20·34-s − 10·41-s + 8·46-s − 12·47-s + 34·49-s + 12·50-s − 16·62-s − 8·64-s + 20·68-s − 12·71-s + 18·73-s − 28·79-s − 20·82-s − 28·89-s + 8·92-s − 24·94-s + ⋯
L(s)  = 1  + 1.41·2-s + 4-s + 3.02·7-s + 4.27·14-s − 16-s + 2.42·17-s + 0.834·23-s + 6/5·25-s + 3.02·28-s − 1.43·31-s − 1.41·32-s + 3.42·34-s − 1.56·41-s + 1.17·46-s − 1.75·47-s + 34/7·49-s + 1.69·50-s − 2.03·62-s − 64-s + 2.42·68-s − 1.42·71-s + 2.10·73-s − 3.15·79-s − 2.20·82-s − 2.96·89-s + 0.834·92-s − 2.47·94-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 419904 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 419904 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(419904\)    =    \(2^{6} \cdot 3^{8}\)
Sign: $1$
Analytic conductor: \(26.7734\)
Root analytic conductor: \(2.27471\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 419904,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(6.155641866\)
\(L(\frac12)\) \(\approx\) \(6.155641866\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - p T + p T^{2} \)
3 \( 1 \)
good5$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.5.a_ag
7$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.7.ai_be
11$C_2^2$ \( 1 - 13 T^{2} + p^{2} T^{4} \) 2.11.a_an
13$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \) 2.13.a_aw
17$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \) 2.17.ak_ch
19$C_2^2$ \( 1 - 37 T^{2} + p^{2} T^{4} \) 2.19.a_abl
23$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.23.ae_by
29$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.29.a_acg
31$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.31.i_da
37$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.37.a_acs
41$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \) 2.41.k_ed
43$C_2^2$ \( 1 + 35 T^{2} + p^{2} T^{4} \) 2.43.a_bj
47$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.47.m_fa
53$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.53.a_aec
59$C_2^2$ \( 1 - 117 T^{2} + p^{2} T^{4} \) 2.59.a_aen
61$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.61.a_w
67$C_2^2$ \( 1 - 125 T^{2} + p^{2} T^{4} \) 2.67.a_aev
71$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.71.m_gw
73$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \) 2.73.as_it
79$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \) 2.79.bc_nq
83$C_2^2$ \( 1 - 150 T^{2} + p^{2} T^{4} \) 2.83.a_afu
89$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \) 2.89.bc_ok
97$C_2$ \( ( 1 - T + p T^{2} )^{2} \) 2.97.ac_hn
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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

−11.05676370223419007399706452406, −10.56403900665787677815629779259, −10.03853164089875178232434891928, −9.546120881916243549839023310555, −8.821791370928858914778684800658, −8.430571003450487946054226638599, −8.213069451344319068318173214662, −7.64329466748602418269165683526, −7.16912319377247514927321894021, −6.90467503324176727115680453932, −5.91366289011777225508440899350, −5.55590599032313466479475713771, −5.12522262100955812461867238592, −4.92125145007063426484753044798, −4.49006272121340995238732653547, −3.77047052516372296380308774509, −3.25282451476654786510245084972, −2.62773864332262740372489384822, −1.48808577906672131281369178833, −1.48659727631433953645201304522, 1.48659727631433953645201304522, 1.48808577906672131281369178833, 2.62773864332262740372489384822, 3.25282451476654786510245084972, 3.77047052516372296380308774509, 4.49006272121340995238732653547, 4.92125145007063426484753044798, 5.12522262100955812461867238592, 5.55590599032313466479475713771, 5.91366289011777225508440899350, 6.90467503324176727115680453932, 7.16912319377247514927321894021, 7.64329466748602418269165683526, 8.213069451344319068318173214662, 8.430571003450487946054226638599, 8.821791370928858914778684800658, 9.546120881916243549839023310555, 10.03853164089875178232434891928, 10.56403900665787677815629779259, 11.05676370223419007399706452406

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