↑

Properties

Label 4-864e2-1.1-c1e2-0-14
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

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 6·7-s − 6·13-s + 6·19-s − 6·25-s + 14·37-s + 24·43-s + 13·49-s − 2·61-s + 6·67-s − 30·73-s − 18·79-s + 36·91-s + 18·97-s − 30·103-s + 12·109-s + 14·121-s + 127-s + 131-s − 36·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 169-s + ⋯
L(s)  = 1  − 2.26·7-s − 1.66·13-s + 1.37·19-s − 6/5·25-s + 2.30·37-s + 3.65·43-s + 13/7·49-s − 0.256·61-s + 0.733·67-s − 3.51·73-s − 2.02·79-s + 3.77·91-s + 1.82·97-s − 2.95·103-s + 1.14·109-s + 1.27·121-s + 0.0887·127-s + 0.0873·131-s − 3.12·133-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 + 1/13·169-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$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.5.a_g
7$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.7.g_x
11$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.11.a_ao
13$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.13.g_bj
17$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.17.a_be
19$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.19.ag_bv
23$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.23.a_k
29$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.29.a_ag
31$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.31.a_ck
37$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \) 2.37.ao_et
41$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.41.a_s
43$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \) 2.43.ay_iw
47$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.47.a_cg
53$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.53.a_dm
59$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.59.a_de
61$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.61.c_et
67$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.67.ag_fn
71$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.71.a_ac
73$C_2$ \( ( 1 + 15 T + p T^{2} )^{2} \) 2.73.be_oh
79$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \) 2.79.s_jf
83$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.83.a_w
89$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.89.a_da
97$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \) 2.97.as_kp
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

−7.86400018758168335012851109883, −7.42468008636998742776243725148, −7.28999576050073652042194900023, −6.84703029813025683919349357796, −5.96701174527234514935331354680, −5.96035170554403781150034405946, −5.63427018232312358528631119727, −4.62403138193129190420684686156, −4.36568585015197304210716831899, −3.73252772900173333221011269732, −2.94454619082960172956967045225, −2.88076148445155716360589692436, −2.20258441520663841153734800735, −0.920256988552060674036203635836, 0, 0.920256988552060674036203635836, 2.20258441520663841153734800735, 2.88076148445155716360589692436, 2.94454619082960172956967045225, 3.73252772900173333221011269732, 4.36568585015197304210716831899, 4.62403138193129190420684686156, 5.63427018232312358528631119727, 5.96035170554403781150034405946, 5.96701174527234514935331354680, 6.84703029813025683919349357796, 7.28999576050073652042194900023, 7.42468008636998742776243725148, 7.86400018758168335012851109883

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