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Properties

Label 4-864e2-1.1-c1e2-0-43
Degree $4$
Conductor $746496$
Sign $-1$
Analytic cond. $47.5972$
Root an. cond. $2.62660$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $1$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 6·11-s − 4·19-s − 25-s − 12·41-s + 20·43-s − 13·49-s − 24·59-s − 28·67-s − 14·73-s + 6·83-s − 36·89-s − 2·97-s − 18·107-s − 12·113-s + 5·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 10·169-s + 173-s + 179-s + ⋯
L(s)  = 1  + 1.80·11-s − 0.917·19-s − 1/5·25-s − 1.87·41-s + 3.04·43-s − 1.85·49-s − 3.12·59-s − 3.42·67-s − 1.63·73-s + 0.658·83-s − 3.81·89-s − 0.203·97-s − 1.74·107-s − 1.12·113-s + 5/11·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 0.769·169-s + 0.0760·173-s + 0.0747·179-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 746496 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 746496 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(746496\)    =    \(2^{10} \cdot 3^{6}\)
Sign: $-1$
Analytic conductor: \(47.5972\)
Root analytic conductor: \(2.62660\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 746496,\ (\ :1/2, 1/2),\ -1)\)

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 \( 1 \)
3 \( 1 \)
good5$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.5.a_b
7$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) 2.7.a_n
11$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.11.ag_bf
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.13.a_k
17$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.17.a_bi
19$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.19.e_bq
23$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.23.a_k
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.29.a_w
31$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.31.a_bl
37$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.37.a_cs
41$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.41.m_eo
43$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.43.au_he
47$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.47.a_cg
53$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.53.a_z
59$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.59.y_kc
61$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.61.a_cg
67$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \) 2.67.bc_ms
71$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.71.a_fm
73$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \) 2.73.o_hn
79$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.79.a_dq
83$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.83.ag_gt
89$C_2$ \( ( 1 + 18 T + p T^{2} )^{2} \) 2.89.bk_ti
97$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.97.c_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

−8.104088285674841823256361524017, −7.46895104625204893367477999066, −7.29929175763847648434390847974, −6.47796475592373752838072674203, −6.43760550649778711412630154056, −5.89467970712550736408970851242, −5.43287815412844995352588032205, −4.47356450228848094997188248522, −4.43365380485938957073214377677, −3.92407328889816322387260499130, −3.12927788056823217499419163319, −2.78595567768432254175608596610, −1.56286118008502235583804201463, −1.52887349549650490954756981970, 0, 1.52887349549650490954756981970, 1.56286118008502235583804201463, 2.78595567768432254175608596610, 3.12927788056823217499419163319, 3.92407328889816322387260499130, 4.43365380485938957073214377677, 4.47356450228848094997188248522, 5.43287815412844995352588032205, 5.89467970712550736408970851242, 6.43760550649778711412630154056, 6.47796475592373752838072674203, 7.29929175763847648434390847974, 7.46895104625204893367477999066, 8.104088285674841823256361524017

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