Properties

Label 4-3200e2-1.1-c1e2-0-23
Degree $4$
Conductor $10240000$
Sign $1$
Analytic cond. $652.911$
Root an. cond. $5.05491$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 6·9-s − 8·13-s + 24·37-s + 20·41-s + 14·49-s + 8·53-s + 27·81-s + 20·89-s + 48·117-s + 22·121-s + ⋯
L(s)  = 1  − 2·9-s − 2.21·13-s + 3.94·37-s + 3.12·41-s + 2·49-s + 1.09·53-s + 3·81-s + 2.11·89-s + 4.43·117-s + 2·121-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 10240000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10240000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(10240000\)    =    \(2^{14} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(652.911\)
Root analytic conductor: \(5.05491\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 10240000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.944596205\)
\(L(\frac12)\) \(\approx\) \(1.944596205\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 \)
good3$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.3.a_g
7$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.7.a_ao
11$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.11.a_aw
13$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.13.i_bq
17$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.17.a_abe
19$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.19.a_abm
23$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.23.a_abu
29$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.29.a_abq
31$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.31.a_ck
37$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \) 2.37.ay_ik
41$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.41.au_ha
43$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.43.a_di
47$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.47.a_adq
53$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.53.ai_es
59$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.59.a_aeo
61$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.61.a_w
67$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.67.a_fe
71$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.71.a_fm
73$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) 2.73.a_aeg
79$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.79.a_gc
83$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.83.a_gk
89$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.89.au_ks
97$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.97.a_fa
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

−8.864229234684621713217378287232, −8.566839174917558726316243266225, −7.896149526479872609028048004255, −7.80852877735676114772836925711, −7.39303259468716917458983305383, −7.25438679845608531451070043837, −6.38193377788816080786625430517, −6.21341210768247059980174048170, −5.70510013203006760095319576936, −5.65653825307123333157078792124, −4.92256337003355570242357659489, −4.76958668077760820362136380922, −4.06320426163702879380594545379, −3.97484686586189709183783656791, −2.89680729236596222818665924399, −2.84565555392979675698912581105, −2.35382097286369136464435644832, −2.20606770134580424980998439924, −0.76679370213445048722903756590, −0.61710629357465666210310724734, 0.61710629357465666210310724734, 0.76679370213445048722903756590, 2.20606770134580424980998439924, 2.35382097286369136464435644832, 2.84565555392979675698912581105, 2.89680729236596222818665924399, 3.97484686586189709183783656791, 4.06320426163702879380594545379, 4.76958668077760820362136380922, 4.92256337003355570242357659489, 5.65653825307123333157078792124, 5.70510013203006760095319576936, 6.21341210768247059980174048170, 6.38193377788816080786625430517, 7.25438679845608531451070043837, 7.39303259468716917458983305383, 7.80852877735676114772836925711, 7.896149526479872609028048004255, 8.566839174917558726316243266225, 8.864229234684621713217378287232

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