↑

Properties

Label 4-864e2-1.1-c1e2-0-24
Degree $4$
Conductor $746496$
Sign $-1$
Analytic cond. $47.5972$
Root an. cond. $2.62660$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 6·11-s + 5·19-s + 8·25-s − 15·41-s − 7·43-s + 5·49-s − 3·59-s − 10·67-s − 5·73-s + 21·83-s − 18·89-s + 16·97-s − 9·107-s + 12·113-s + 14·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 8·169-s + 173-s + 179-s + ⋯
L(s)  = 1  − 1.80·11-s + 1.14·19-s + 8/5·25-s − 2.34·41-s − 1.06·43-s + 5/7·49-s − 0.390·59-s − 1.22·67-s − 0.585·73-s + 2.30·83-s − 1.90·89-s + 1.62·97-s − 0.870·107-s + 1.12·113-s + 1.27·121-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 + 8/13·169-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: yes
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$C_2^2$ \( 1 - 8 T^{2} + p^{2} T^{4} \) 2.5.a_ai
7$C_2^2$ \( 1 - 5 T^{2} + p^{2} T^{4} \) 2.7.a_af
11$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.11.g_w
13$C_2^2$ \( 1 - 8 T^{2} + p^{2} T^{4} \) 2.13.a_ai
17$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.17.a_z
19$C_2$$\times$$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.19.af_y
23$C_2^2$ \( 1 + T^{2} + p^{2} T^{4} \) 2.23.a_b
29$C_2^2$ \( 1 + 40 T^{2} + p^{2} T^{4} \) 2.29.a_bo
31$C_2^2$ \( 1 + 19 T^{2} + p^{2} T^{4} \) 2.31.a_t
37$C_2^2$ \( 1 + 52 T^{2} + p^{2} T^{4} \) 2.37.a_ca
41$C_2$$\times$$C_2$ \( ( 1 + 6 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.41.p_fg
43$C_2$$\times$$C_2$ \( ( 1 - T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.43.h_da
47$C_2^2$ \( 1 - 59 T^{2} + p^{2} T^{4} \) 2.47.a_ach
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.53.a_cs
59$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.59.d_cm
61$C_2^2$ \( 1 + 112 T^{2} + p^{2} T^{4} \) 2.61.a_ei
67$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.67.k_da
71$C_2$ \( ( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) 2.71.a_adf
73$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.73.f_fc
79$C_2^2$ \( 1 + 103 T^{2} + p^{2} T^{4} \) 2.79.a_dz
83$C_2$$\times$$C_2$ \( ( 1 - 15 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) 2.83.av_jw
89$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 18 T + p T^{2} ) \) 2.89.s_gw
97$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.97.aq_jy
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

−7.994032282415865268137972191957, −7.68286870889962513798219901245, −7.20180902979081824212736940148, −6.79031856578914068460655232230, −6.32843742988877713246246972828, −5.62231283902153284273180036910, −5.26786768646182187811642613514, −4.92410303935684114530511570842, −4.52216724975806029479532901342, −3.58007291442527064245523264577, −3.15847917784775617757409138461, −2.74336165283225855450779555201, −2.00308291990562626754201974288, −1.14802039056275806991843916665, 0, 1.14802039056275806991843916665, 2.00308291990562626754201974288, 2.74336165283225855450779555201, 3.15847917784775617757409138461, 3.58007291442527064245523264577, 4.52216724975806029479532901342, 4.92410303935684114530511570842, 5.26786768646182187811642613514, 5.62231283902153284273180036910, 6.32843742988877713246246972828, 6.79031856578914068460655232230, 7.20180902979081824212736940148, 7.68286870889962513798219901245, 7.994032282415865268137972191957

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