Properties

Label 8-55e8-1.1-c1e4-0-5
Degree $8$
Conductor $8.373\times 10^{13}$
Sign $1$
Analytic cond. $340415.$
Root an. cond. $4.91474$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $4$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·4-s − 8·9-s + 7·16-s + 4·19-s − 28·31-s + 32·36-s − 16·49-s − 12·59-s − 40·61-s − 8·64-s − 12·71-s − 16·76-s − 8·79-s + 33·81-s − 12·89-s + 36·101-s + 16·109-s + 112·124-s + 127-s + 131-s + 137-s + 139-s − 56·144-s + 149-s + 151-s + 157-s + 163-s + ⋯
L(s)  = 1  − 2·4-s − 8/3·9-s + 7/4·16-s + 0.917·19-s − 5.02·31-s + 16/3·36-s − 2.28·49-s − 1.56·59-s − 5.12·61-s − 64-s − 1.42·71-s − 1.83·76-s − 0.900·79-s + 11/3·81-s − 1.27·89-s + 3.58·101-s + 1.53·109-s + 10.0·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 4.66·144-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(5^{8} \cdot 11^{8}\)
Sign: $1$
Analytic conductor: \(340415.\)
Root analytic conductor: \(4.91474\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{3025} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(4\)
Selberg data: \((8,\ 5^{8} \cdot 11^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

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)$
bad5 \( 1 \)
11 \( 1 \)
good2$D_4\times C_2$ \( 1 + p^{2} T^{2} + 9 T^{4} + p^{4} T^{6} + p^{4} T^{8} \)
3$D_4\times C_2$ \( 1 + 8 T^{2} + 31 T^{4} + 8 p^{2} T^{6} + p^{4} T^{8} \)
7$D_4\times C_2$ \( 1 + 16 T^{2} + 135 T^{4} + 16 p^{2} T^{6} + p^{4} T^{8} \)
13$C_2^2$ \( ( 1 + 8 T^{2} + p^{2} T^{4} )^{2} \)
17$D_4\times C_2$ \( 1 + 52 T^{2} + 1206 T^{4} + 52 p^{2} T^{6} + p^{4} T^{8} \)
19$D_{4}$ \( ( 1 - 2 T + 12 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \)
23$D_4\times C_2$ \( 1 + 40 T^{2} + 1266 T^{4} + 40 p^{2} T^{6} + p^{4} T^{8} \)
29$C_2^2$ \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2} \)
31$D_{4}$ \( ( 1 + 14 T + 108 T^{2} + 14 p T^{3} + p^{2} T^{4} )^{2} \)
37$D_4\times C_2$ \( 1 + 100 T^{2} + 4806 T^{4} + 100 p^{2} T^{6} + p^{4} T^{8} \)
41$C_2^2$ \( ( 1 + 79 T^{2} + p^{2} T^{4} )^{2} \)
43$D_4\times C_2$ \( 1 + 16 T^{2} + 2439 T^{4} + 16 p^{2} T^{6} + p^{4} T^{8} \)
47$D_4\times C_2$ \( 1 + 40 T^{2} + 33 p T^{4} + 40 p^{2} T^{6} + p^{4} T^{8} \)
53$D_4\times C_2$ \( 1 + 64 T^{2} + 1842 T^{4} + 64 p^{2} T^{6} + p^{4} T^{8} \)
59$D_{4}$ \( ( 1 + 6 T + 124 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \)
61$D_{4}$ \( ( 1 + 20 T + 219 T^{2} + 20 p T^{3} + p^{2} T^{4} )^{2} \)
67$D_4\times C_2$ \( 1 + 40 T^{2} - 2529 T^{4} + 40 p^{2} T^{6} + p^{4} T^{8} \)
71$D_{4}$ \( ( 1 + 6 T + 124 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \)
73$C_2^2$ \( ( 1 + 122 T^{2} + p^{2} T^{4} )^{2} \)
79$D_{4}$ \( ( 1 + 4 T + 150 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \)
83$C_2^2$ \( ( 1 + 68 T^{2} + p^{2} T^{4} )^{2} \)
89$D_{4}$ \( ( 1 + 6 T + 175 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \)
97$D_4\times C_2$ \( 1 + 304 T^{2} + 40194 T^{4} + 304 p^{2} T^{6} + p^{4} T^{8} \)
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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

−6.41783765621571101022801522029, −6.08092665150685622626680035033, −6.04718050972638017506347344742, −6.03441570535166690865014845829, −5.68295117238408958911270163393, −5.63211137626596914606913727733, −5.40280877030007592551140254513, −5.13915486301664176180498898473, −4.87065562354408528436738238454, −4.79736981394238779451550480042, −4.55164923330421899238332547875, −4.51088373721475867260674830648, −4.23362150147733966298971941556, −3.64835544944971181545803879043, −3.58220034193358197913409908408, −3.40408663410559302778914222891, −3.38126791470385816451271465428, −3.07364467709568210214719778214, −2.95012040401598536026634142322, −2.38845495692263852713418227376, −2.33119010471559386752444427797, −1.81384883113694686248083067958, −1.70421975595231102591737677079, −1.26508000662824901816968263848, −1.05756869788692709244837772831, 0, 0, 0, 0, 1.05756869788692709244837772831, 1.26508000662824901816968263848, 1.70421975595231102591737677079, 1.81384883113694686248083067958, 2.33119010471559386752444427797, 2.38845495692263852713418227376, 2.95012040401598536026634142322, 3.07364467709568210214719778214, 3.38126791470385816451271465428, 3.40408663410559302778914222891, 3.58220034193358197913409908408, 3.64835544944971181545803879043, 4.23362150147733966298971941556, 4.51088373721475867260674830648, 4.55164923330421899238332547875, 4.79736981394238779451550480042, 4.87065562354408528436738238454, 5.13915486301664176180498898473, 5.40280877030007592551140254513, 5.63211137626596914606913727733, 5.68295117238408958911270163393, 6.03441570535166690865014845829, 6.04718050972638017506347344742, 6.08092665150685622626680035033, 6.41783765621571101022801522029

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