Properties

Label 8-1900e4-1.1-c1e4-0-2
Degree $8$
Conductor $1.303\times 10^{13}$
Sign $1$
Analytic cond. $52981.3$
Root an. cond. $3.89507$
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  + 4·9-s − 4·19-s + 8·31-s − 24·41-s + 20·49-s + 8·61-s − 24·71-s + 16·79-s + 6·81-s + 48·89-s − 24·101-s + 16·109-s − 20·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 44·169-s − 16·171-s + 173-s + 179-s + 181-s + ⋯
L(s)  = 1  + 4/3·9-s − 0.917·19-s + 1.43·31-s − 3.74·41-s + 20/7·49-s + 1.02·61-s − 2.84·71-s + 1.80·79-s + 2/3·81-s + 5.08·89-s − 2.38·101-s + 1.53·109-s − 1.81·121-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 + 3.38·169-s − 1.22·171-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{8} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{8} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{8} \cdot 5^{8} \cdot 19^{4}\)
Sign: $1$
Analytic conductor: \(52981.3\)
Root analytic conductor: \(3.89507\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1900} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{8} \cdot 5^{8} \cdot 19^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(1.728572008\)
\(L(\frac12)\) \(\approx\) \(1.728572008\)
\(L(\frac{3}{2})\) 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)$
bad2 \( 1 \)
5 \( 1 \)
19$C_1$ \( ( 1 + T )^{4} \)
good3$D_4\times C_2$ \( 1 - 4 T^{2} + 10 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} \)
7$C_2^2$ \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \)
11$C_2^2$ \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2} \)
13$D_4\times C_2$ \( 1 - 44 T^{2} + 810 T^{4} - 44 p^{2} T^{6} + p^{4} T^{8} \)
17$C_2^2$ \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \)
23$C_2^2$ \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{2} \)
29$C_2^2$ \( ( 1 + 46 T^{2} + p^{2} T^{4} )^{2} \)
31$D_{4}$ \( ( 1 - 4 T + 54 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \)
37$D_4\times C_2$ \( 1 - 92 T^{2} + 4554 T^{4} - 92 p^{2} T^{6} + p^{4} T^{8} \)
41$C_2$ \( ( 1 + 6 T + p T^{2} )^{4} \)
43$D_4\times C_2$ \( 1 - 68 T^{2} + 4086 T^{4} - 68 p^{2} T^{6} + p^{4} T^{8} \)
47$D_4\times C_2$ \( 1 - 20 T^{2} - 2394 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \)
53$D_4\times C_2$ \( 1 - 44 T^{2} + 5130 T^{4} - 44 p^{2} T^{6} + p^{4} T^{8} \)
59$C_2^2$ \( ( 1 + 70 T^{2} + p^{2} T^{4} )^{2} \)
61$D_{4}$ \( ( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \)
67$D_4\times C_2$ \( 1 - 212 T^{2} + 19914 T^{4} - 212 p^{2} T^{6} + p^{4} T^{8} \)
71$D_{4}$ \( ( 1 + 12 T + 166 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \)
73$D_4\times C_2$ \( 1 - 236 T^{2} + 23814 T^{4} - 236 p^{2} T^{6} + p^{4} T^{8} \)
79$D_{4}$ \( ( 1 - 8 T + 126 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \)
83$C_2^2$ \( ( 1 - 154 T^{2} + p^{2} T^{4} )^{2} \)
89$D_{4}$ \( ( 1 - 24 T + 310 T^{2} - 24 p T^{3} + p^{2} T^{4} )^{2} \)
97$D_4\times C_2$ \( 1 + 100 T^{2} + 20346 T^{4} + 100 p^{2} T^{6} + p^{4} T^{8} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−6.65122693251984489817794257360, −6.28647529078127390787772412907, −6.22584884157680453077185216869, −6.13489504513085329214374585161, −5.58943526163631367155100796194, −5.45895404721001998588141621011, −5.24693946134671104632165963438, −5.07764536967559769974554854520, −4.81459529214906103461230816430, −4.62423123865531631987689892022, −4.36732104761944392557927208307, −4.18555733333832005450242237785, −3.94757589520230145568180429970, −3.65443757765760479135605192137, −3.54608357388191831612022758817, −3.20248208932546301336539203707, −3.04697598017794262047488864882, −2.43818718212792993491526614566, −2.32410381523485315122514870649, −2.28078474052169429511839132782, −1.81405962484030037348432388826, −1.35632934320536687730304894410, −1.26417365099105297250291928728, −0.860208145605935195256676104336, −0.23667438150447221158205953482, 0.23667438150447221158205953482, 0.860208145605935195256676104336, 1.26417365099105297250291928728, 1.35632934320536687730304894410, 1.81405962484030037348432388826, 2.28078474052169429511839132782, 2.32410381523485315122514870649, 2.43818718212792993491526614566, 3.04697598017794262047488864882, 3.20248208932546301336539203707, 3.54608357388191831612022758817, 3.65443757765760479135605192137, 3.94757589520230145568180429970, 4.18555733333832005450242237785, 4.36732104761944392557927208307, 4.62423123865531631987689892022, 4.81459529214906103461230816430, 5.07764536967559769974554854520, 5.24693946134671104632165963438, 5.45895404721001998588141621011, 5.58943526163631367155100796194, 6.13489504513085329214374585161, 6.22584884157680453077185216869, 6.28647529078127390787772412907, 6.65122693251984489817794257360

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