Properties

Label 8-350e4-1.1-c1e4-0-1
Degree $8$
Conductor $15006250000$
Sign $1$
Analytic cond. $61.0071$
Root an. cond. $1.67175$
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-s − 6·9-s + 4·11-s − 24·29-s + 14·31-s − 6·36-s − 28·41-s + 4·44-s + 13·49-s − 28·59-s + 28·61-s − 64-s − 4·71-s − 22·79-s + 9·81-s + 14·89-s − 24·99-s + 24·109-s − 24·116-s + 26·121-s + 14·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯
L(s)  = 1  + 1/2·4-s − 2·9-s + 1.20·11-s − 4.45·29-s + 2.51·31-s − 36-s − 4.37·41-s + 0.603·44-s + 13/7·49-s − 3.64·59-s + 3.58·61-s − 1/8·64-s − 0.474·71-s − 2.47·79-s + 81-s + 1.48·89-s − 2.41·99-s + 2.29·109-s − 2.22·116-s + 2.36·121-s + 1.25·124-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 + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 5^{8} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(61.0071\)
Root analytic conductor: \(1.67175\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{4} \cdot 5^{8} \cdot 7^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(1.134997821\)
\(L(\frac12)\) \(\approx\) \(1.134997821\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 - T^{2} + T^{4} \)
5 \( 1 \)
7$C_2^2$ \( 1 - 13 T^{2} + p^{2} T^{4} \)
good3$C_2^2$ \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \)
11$C_2^2$ \( ( 1 - 2 T - 7 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \)
13$C_2$ \( ( 1 - p T^{2} )^{4} \)
17$C_2^3$ \( 1 - 15 T^{2} - 64 T^{4} - 15 p^{2} T^{6} + p^{4} T^{8} \)
19$C_2^2$ \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \)
23$C_2^3$ \( 1 + 37 T^{2} + 840 T^{4} + 37 p^{2} T^{6} + p^{4} T^{8} \)
29$C_2$ \( ( 1 + 6 T + p T^{2} )^{4} \)
31$C_2$ \( ( 1 - 11 T + p T^{2} )^{2}( 1 + 4 T + p T^{2} )^{2} \)
37$C_2^3$ \( 1 + 58 T^{2} + 1995 T^{4} + 58 p^{2} T^{6} + p^{4} T^{8} \)
41$C_2$ \( ( 1 + 7 T + p T^{2} )^{4} \)
43$C_2^2$ \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \)
47$C_2^3$ \( 1 + 45 T^{2} - 184 T^{4} + 45 p^{2} T^{6} + p^{4} T^{8} \)
53$C_2^2$$\times$$C_2^2$ \( ( 1 - 14 T + 143 T^{2} - 14 p T^{3} + p^{2} T^{4} )( 1 + 14 T + 143 T^{2} + 14 p T^{3} + p^{2} T^{4} ) \)
59$C_2^2$ \( ( 1 + 14 T + 137 T^{2} + 14 p T^{3} + p^{2} T^{4} )^{2} \)
61$C_2$ \( ( 1 - 13 T + p T^{2} )^{2}( 1 - T + p T^{2} )^{2} \)
67$C_2^3$ \( 1 - 10 T^{2} - 4389 T^{4} - 10 p^{2} T^{6} + p^{4} T^{8} \)
71$C_2$ \( ( 1 + T + p T^{2} )^{4} \)
73$C_2^3$ \( 1 - 50 T^{2} - 2829 T^{4} - 50 p^{2} T^{6} + p^{4} T^{8} \)
79$C_2^2$ \( ( 1 + 11 T + 42 T^{2} + 11 p T^{3} + p^{2} T^{4} )^{2} \)
83$C_2^2$ \( ( 1 + 30 T^{2} + p^{2} T^{4} )^{2} \)
89$C_2^2$ \( ( 1 - 7 T - 40 T^{2} - 7 p T^{3} + p^{2} T^{4} )^{2} \)
97$C_2^2$ \( ( 1 - 145 T^{2} + p^{2} T^{4} )^{2} \)
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

−8.496063414300731290279259296454, −8.204297354555299854805385781259, −7.77570836985041152075369572598, −7.64170337091796259960361270332, −7.27072997061889653830996042287, −7.03464507502967990954147533753, −6.89638452408749512590416418684, −6.54248515544569018388702395408, −6.15513546850204214880593903738, −6.15150914836401806077472135918, −5.67486444821599156887001019560, −5.53576678643747861766465223510, −5.42021458664854185025725445700, −4.90789011547799026148635882731, −4.68804610473596988564689145623, −4.16866772115452463122730823764, −4.02322756502015697176893815719, −3.51770189566314867206876180782, −3.26687968000425660613001941969, −3.13890896507679599036655756324, −2.68797973762886446199521333512, −2.00891221474126650298704887708, −1.96514011204830873215476330558, −1.44984773523033436262610638422, −0.42669489621446936658380505525, 0.42669489621446936658380505525, 1.44984773523033436262610638422, 1.96514011204830873215476330558, 2.00891221474126650298704887708, 2.68797973762886446199521333512, 3.13890896507679599036655756324, 3.26687968000425660613001941969, 3.51770189566314867206876180782, 4.02322756502015697176893815719, 4.16866772115452463122730823764, 4.68804610473596988564689145623, 4.90789011547799026148635882731, 5.42021458664854185025725445700, 5.53576678643747861766465223510, 5.67486444821599156887001019560, 6.15150914836401806077472135918, 6.15513546850204214880593903738, 6.54248515544569018388702395408, 6.89638452408749512590416418684, 7.03464507502967990954147533753, 7.27072997061889653830996042287, 7.64170337091796259960361270332, 7.77570836985041152075369572598, 8.204297354555299854805385781259, 8.496063414300731290279259296454

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