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Properties

Label 4-1856e2-1.1-c1e2-0-0
Degree $4$
Conductor $3444736$
Sign $1$
Analytic cond. $219.639$
Root an. cond. $3.84970$
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  + 6·5-s − 4·7-s + 9-s + 2·13-s − 12·23-s + 17·25-s + 6·29-s − 24·35-s + 6·45-s − 2·49-s + 18·53-s + 12·59-s − 4·63-s + 12·65-s + 16·67-s − 8·81-s − 12·83-s − 8·91-s + 8·103-s + 36·107-s − 10·109-s − 72·115-s + 2·117-s + 17·121-s + 18·125-s + 127-s + 131-s + ⋯
L(s)  = 1  + 2.68·5-s − 1.51·7-s + 1/3·9-s + 0.554·13-s − 2.50·23-s + 17/5·25-s + 1.11·29-s − 4.05·35-s + 0.894·45-s − 2/7·49-s + 2.47·53-s + 1.56·59-s − 0.503·63-s + 1.48·65-s + 1.95·67-s − 8/9·81-s − 1.31·83-s − 0.838·91-s + 0.788·103-s + 3.48·107-s − 0.957·109-s − 6.71·115-s + 0.184·117-s + 1.54·121-s + 1.60·125-s + 0.0887·127-s + 0.0873·131-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3444736 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3444736 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(3444736\)    =    \(2^{12} \cdot 29^{2}\)
Sign: $1$
Analytic conductor: \(219.639\)
Root analytic conductor: \(3.84970\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 3444736,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.384006701\)
\(L(\frac12)\) \(\approx\) \(3.384006701\)
\(L(\frac{3}{2})\) 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 \( 1 \)
29$C_2$ \( 1 - 6 T + p T^{2} \)
good3$C_2^2$ \( 1 - T^{2} + p^{2} T^{4} \) 2.3.a_ab
5$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.5.ag_t
7$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.7.e_s
11$C_2^2$ \( 1 - 17 T^{2} + p^{2} T^{4} \) 2.11.a_ar
13$C_2$ \( ( 1 - T + p T^{2} )^{2} \) 2.13.ac_bb
17$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \) 2.17.a_ao
19$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.19.a_abm
23$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.23.m_de
31$C_2^2$ \( 1 - 17 T^{2} + p^{2} T^{4} \) 2.31.a_ar
37$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.37.a_acw
41$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.41.a_ack
43$C_2^2$ \( 1 - 41 T^{2} + p^{2} T^{4} \) 2.43.a_abp
47$C_2^2$ \( 1 - 89 T^{2} + p^{2} T^{4} \) 2.47.a_adl
53$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \) 2.53.as_hf
59$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.59.am_fy
61$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.61.a_cg
67$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.67.aq_hq
71$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.71.a_fm
73$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.73.a_afq
79$C_2^2$ \( 1 - 113 T^{2} + p^{2} T^{4} \) 2.79.a_aej
83$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.83.m_hu
89$C_2^2$ \( 1 - 158 T^{2} + p^{2} T^{4} \) 2.89.a_agc
97$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \) 2.97.a_ao
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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

−9.718401207174014503172660308893, −9.042285192399608718153223202057, −8.891608137321788966076522715303, −8.337639359831187560210062794260, −8.034641470385678920492917201201, −7.24842491198480076559381025712, −6.88168922737932766630796569940, −6.53452948088980873338242776023, −6.04657607767764937239700105556, −6.04463265710505315401007838528, −5.62079058345781217666466243667, −5.18847466938266158498358514536, −4.57954744281582734532647341384, −3.87378433375589717569562278782, −3.66161620111455688895882748141, −2.87200808338468111820701320434, −2.40323605185343480024635260788, −2.05622677713784741169377606313, −1.52432755453134803901369365234, −0.66310018716206193729112951394, 0.66310018716206193729112951394, 1.52432755453134803901369365234, 2.05622677713784741169377606313, 2.40323605185343480024635260788, 2.87200808338468111820701320434, 3.66161620111455688895882748141, 3.87378433375589717569562278782, 4.57954744281582734532647341384, 5.18847466938266158498358514536, 5.62079058345781217666466243667, 6.04463265710505315401007838528, 6.04657607767764937239700105556, 6.53452948088980873338242776023, 6.88168922737932766630796569940, 7.24842491198480076559381025712, 8.034641470385678920492917201201, 8.337639359831187560210062794260, 8.891608137321788966076522715303, 9.042285192399608718153223202057, 9.718401207174014503172660308893

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