Properties

Label 8-50e8-1.1-c0e4-0-1
Degree $8$
Conductor $3.906\times 10^{13}$
Sign $1$
Analytic cond. $2.42319$
Root an. cond. $1.11698$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·4-s − 4·9-s + 3·16-s + 2·29-s + 8·36-s − 2·41-s − 4·49-s − 2·61-s − 4·64-s + 10·81-s + 2·89-s − 2·101-s + 2·109-s − 4·116-s + 4·121-s + 127-s + 131-s + 137-s + 139-s − 12·144-s + 149-s + 151-s + 157-s + 163-s + 4·164-s + 167-s + 169-s + ⋯
L(s)  = 1  − 2·4-s − 4·9-s + 3·16-s + 2·29-s + 8·36-s − 2·41-s − 4·49-s − 2·61-s − 4·64-s + 10·81-s + 2·89-s − 2·101-s + 2·109-s − 4·116-s + 4·121-s + 127-s + 131-s + 137-s + 139-s − 12·144-s + 149-s + 151-s + 157-s + 163-s + 4·164-s + 167-s + 169-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{8} \cdot 5^{16}\)
Sign: $1$
Analytic conductor: \(2.42319\)
Root analytic conductor: \(1.11698\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{2500} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{8} \cdot 5^{16} ,\ ( \ : 0, 0, 0, 0 ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2910836402\)
\(L(\frac12)\) \(\approx\) \(0.2910836402\)
\(L(1)\) 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$C_2$ \( ( 1 + T^{2} )^{2} \)
5 \( 1 \)
good3$C_2$ \( ( 1 + T^{2} )^{4} \)
7$C_2$ \( ( 1 + T^{2} )^{4} \)
11$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
13$C_4\times C_2$ \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \)
17$C_4\times C_2$ \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \)
19$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
23$C_2$ \( ( 1 + T^{2} )^{4} \)
29$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \)
31$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
37$C_4\times C_2$ \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \)
41$C_4$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
43$C_2$ \( ( 1 + T^{2} )^{4} \)
47$C_2$ \( ( 1 + T^{2} )^{4} \)
53$C_4\times C_2$ \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \)
59$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
61$C_4$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
67$C_2$ \( ( 1 + T^{2} )^{4} \)
71$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
73$C_4\times C_2$ \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \)
79$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
83$C_2$ \( ( 1 + T^{2} )^{4} \)
89$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \)
97$C_4\times C_2$ \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−6.41578702818397139456434879587, −6.14759707981169934111053193586, −6.01201859067437568955349665841, −5.94066651665368945124182990021, −5.77882255759706182447512753027, −5.38646890034592018374366773567, −5.28402419773630207685451165134, −4.97651031618491700260342894964, −4.86412263691690495300321104041, −4.80072268899660433167958533506, −4.65568841957767370800265542642, −4.18288672127242375762140413149, −4.10149693836474116741102207030, −3.46840180422705486251243593380, −3.44772444460552869119794460845, −3.27777613987017360413116688629, −3.13289193927527128182397557589, −3.04799198685759782132689532728, −2.60856274578672415848975273332, −2.43083820172957114867873953669, −1.94002225091826912362541592745, −1.67676929632192092176702907719, −1.32037444264783487904565903411, −0.62828097749147127048723355992, −0.40955603311560012748092629782, 0.40955603311560012748092629782, 0.62828097749147127048723355992, 1.32037444264783487904565903411, 1.67676929632192092176702907719, 1.94002225091826912362541592745, 2.43083820172957114867873953669, 2.60856274578672415848975273332, 3.04799198685759782132689532728, 3.13289193927527128182397557589, 3.27777613987017360413116688629, 3.44772444460552869119794460845, 3.46840180422705486251243593380, 4.10149693836474116741102207030, 4.18288672127242375762140413149, 4.65568841957767370800265542642, 4.80072268899660433167958533506, 4.86412263691690495300321104041, 4.97651031618491700260342894964, 5.28402419773630207685451165134, 5.38646890034592018374366773567, 5.77882255759706182447512753027, 5.94066651665368945124182990021, 6.01201859067437568955349665841, 6.14759707981169934111053193586, 6.41578702818397139456434879587

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