↑

Properties

Label 4-2178e2-1.1-c1e2-0-10
Degree $4$
Conductor $4743684$
Sign $1$
Analytic cond. $302.461$
Root an. cond. $4.17030$
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  + 2·2-s + 3·4-s + 4·8-s + 5·16-s + 12·17-s + 2·25-s − 12·29-s − 16·31-s + 6·32-s + 24·34-s − 4·37-s + 12·41-s − 2·49-s + 4·50-s − 24·58-s − 32·62-s + 7·64-s + 8·67-s + 36·68-s − 8·74-s + 24·82-s + 24·83-s − 20·97-s − 4·98-s + 6·100-s + 36·101-s − 32·103-s + ⋯
L(s)  = 1  + 1.41·2-s + 3/2·4-s + 1.41·8-s + 5/4·16-s + 2.91·17-s + 2/5·25-s − 2.22·29-s − 2.87·31-s + 1.06·32-s + 4.11·34-s − 0.657·37-s + 1.87·41-s − 2/7·49-s + 0.565·50-s − 3.15·58-s − 4.06·62-s + 7/8·64-s + 0.977·67-s + 4.36·68-s − 0.929·74-s + 2.65·82-s + 2.63·83-s − 2.03·97-s − 0.404·98-s + 3/5·100-s + 3.58·101-s − 3.15·103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4743684 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4743684 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(4743684\)    =    \(2^{2} \cdot 3^{4} \cdot 11^{4}\)
Sign: $1$
Analytic conductor: \(302.461\)
Root analytic conductor: \(4.17030\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 4743684,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(6.965042238\)
\(L(\frac12)\) \(\approx\) \(6.965042238\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{2} \)
3 \( 1 \)
11 \( 1 \)
good5$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.5.a_ac
7$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.7.a_c
13$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \) 2.13.a_o
17$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.17.am_cs
19$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.19.a_ak
23$C_2^2$ \( 1 + 34 T^{2} + p^{2} T^{4} \) 2.23.a_bi
29$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.29.m_dq
31$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.31.q_ew
37$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.37.e_da
41$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.41.am_eo
43$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.43.a_di
47$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \) 2.47.a_ao
53$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.53.a_ac
59$C_2^2$ \( 1 + 70 T^{2} + p^{2} T^{4} \) 2.59.a_cs
61$C_2^2$ \( 1 + 110 T^{2} + p^{2} T^{4} \) 2.61.a_eg
67$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.67.ai_fu
71$C_2^2$ \( 1 + 130 T^{2} + p^{2} T^{4} \) 2.71.a_fa
73$C_2^2$ \( 1 - 46 T^{2} + p^{2} T^{4} \) 2.73.a_abu
79$C_2^2$ \( 1 + 146 T^{2} + p^{2} T^{4} \) 2.79.a_fq
83$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \) 2.83.ay_ly
89$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \) 2.89.a_ao
97$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.97.u_li
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

−9.218846647266054357255285169383, −9.094150986013175514922413777611, −8.223850184439115354673537147746, −7.966709082705430701772960447128, −7.52633410466568362151748159832, −7.27417172720209469535895384438, −7.05597760164910291562413115026, −6.33648496466454407142480253979, −5.84678309940390184304269883176, −5.66627082218340301688879432238, −5.21913843494221814288090726907, −5.16191686503733930132580416292, −4.31965283118642362147198683925, −3.80947446600837650140939222033, −3.56372967774523466564750257836, −3.28694517513706333517940938019, −2.65340378767365384133327502168, −1.91625927193738007672887048928, −1.61222589263046787591331898516, −0.72395614243674917297345015033, 0.72395614243674917297345015033, 1.61222589263046787591331898516, 1.91625927193738007672887048928, 2.65340378767365384133327502168, 3.28694517513706333517940938019, 3.56372967774523466564750257836, 3.80947446600837650140939222033, 4.31965283118642362147198683925, 5.16191686503733930132580416292, 5.21913843494221814288090726907, 5.66627082218340301688879432238, 5.84678309940390184304269883176, 6.33648496466454407142480253979, 7.05597760164910291562413115026, 7.27417172720209469535895384438, 7.52633410466568362151748159832, 7.966709082705430701772960447128, 8.223850184439115354673537147746, 9.094150986013175514922413777611, 9.218846647266054357255285169383

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