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Properties

Label 4-2178e2-1.1-c1e2-0-32
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 $2$

Origins

Origins of factors

Downloads

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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: \(2\)
Selberg data: \((4,\ 4743684,\ (\ :1/2, 1/2),\ 1)\)

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$C_1$ \( ( 1 + T )^{2} \)
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.m_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.am_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.m_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.y_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
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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

−9.045988152678817840719674687896, −8.414051045401048384443744734986, −8.312552229113317140821571033089, −7.974118909830113277898075007181, −7.17674159499947426540572613454, −6.84570371632624073085198253387, −6.75578396103136766201724422059, −6.58787292885126702103827837214, −5.67920969573548208024374822969, −5.47094007033801386776092476236, −4.89405740597908458181000854011, −4.36076629803567189039731581546, −3.95677079647513392939892075495, −3.35902641318813610654946834557, −2.56169948959042292204830253299, −2.53889350872144173817929234141, −1.64980167632186299370178743758, −1.39634000930191349015784829104, 0, 0, 1.39634000930191349015784829104, 1.64980167632186299370178743758, 2.53889350872144173817929234141, 2.56169948959042292204830253299, 3.35902641318813610654946834557, 3.95677079647513392939892075495, 4.36076629803567189039731581546, 4.89405740597908458181000854011, 5.47094007033801386776092476236, 5.67920969573548208024374822969, 6.58787292885126702103827837214, 6.75578396103136766201724422059, 6.84570371632624073085198253387, 7.17674159499947426540572613454, 7.974118909830113277898075007181, 8.312552229113317140821571033089, 8.414051045401048384443744734986, 9.045988152678817840719674687896

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