↑

Properties

Label 4-864e2-1.1-c1e2-0-27
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  − 2·7-s + 2·13-s + 10·19-s − 6·25-s − 16·31-s − 10·37-s + 8·43-s − 11·49-s + 6·61-s + 26·67-s + 18·73-s − 22·79-s − 4·91-s + 2·97-s + 6·103-s − 20·109-s − 18·121-s + 127-s + 131-s − 20·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯
L(s)  = 1  − 0.755·7-s + 0.554·13-s + 2.29·19-s − 6/5·25-s − 2.87·31-s − 1.64·37-s + 1.21·43-s − 1.57·49-s + 0.768·61-s + 3.17·67-s + 2.10·73-s − 2.47·79-s − 0.419·91-s + 0.203·97-s + 0.591·103-s − 1.91·109-s − 1.63·121-s + 0.0887·127-s + 0.0873·131-s − 1.73·133-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 + ⋯

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$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.7.c_p
11$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.11.a_s
13$C_2$ \( ( 1 - T + p T^{2} )^{2} \) 2.13.ac_bb
17$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.17.a_ac
19$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \) 2.19.ak_cl
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$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.31.q_ew
37$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \) 2.37.k_dv
41$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.41.a_s
43$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.43.ai_dy
47$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.47.a_ag
53$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.53.a_dm
59$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.59.a_ada
61$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.61.ag_fb
67$C_2$ \( ( 1 - 13 T + p T^{2} )^{2} \) 2.67.aba_lr
71$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.71.a_ew
73$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \) 2.73.as_it
79$C_2$ \( ( 1 + 11 T + p T^{2} )^{2} \) 2.79.w_kt
83$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.83.a_w
89$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.89.a_gs
97$C_2$ \( ( 1 - T + p T^{2} )^{2} \) 2.97.ac_hn
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

−7.972335257658859713952381664704, −7.57195972780640499908950962067, −7.23561788630982287928838202899, −6.71110929993388671275045148454, −6.36176902503839563041243175150, −5.61589452328752483822361023345, −5.31494333285431893846186259722, −5.17169676997140923385106697279, −4.05174927146180313890169025811, −3.58529057716851715664298339034, −3.50900323268563987471302746975, −2.66421157652505017015711163951, −1.91159340940988138724333401775, −1.19497678164755144713673653335, 0, 1.19497678164755144713673653335, 1.91159340940988138724333401775, 2.66421157652505017015711163951, 3.50900323268563987471302746975, 3.58529057716851715664298339034, 4.05174927146180313890169025811, 5.17169676997140923385106697279, 5.31494333285431893846186259722, 5.61589452328752483822361023345, 6.36176902503839563041243175150, 6.71110929993388671275045148454, 7.23561788630982287928838202899, 7.57195972780640499908950962067, 7.972335257658859713952381664704

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