Properties

Label 8-2760e4-1.1-c0e4-0-1
Degree $8$
Conductor $5.803\times 10^{13}$
Sign $1$
Analytic cond. $3.59968$
Root an. cond. $1.17363$
Motivic weight $0$
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  + 4·3-s + 10·9-s + 4·13-s + 20·27-s − 4·29-s − 4·31-s + 16·39-s + 4·59-s + 35·81-s − 16·87-s − 16·93-s + 40·117-s − 4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 8·169-s + 173-s + 16·177-s + 179-s + 181-s + ⋯
L(s)  = 1  + 4·3-s + 10·9-s + 4·13-s + 20·27-s − 4·29-s − 4·31-s + 16·39-s + 4·59-s + 35·81-s − 16·87-s − 16·93-s + 40·117-s − 4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 8·169-s + 173-s + 16·177-s + 179-s + 181-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{4} \cdot 5^{4} \cdot 23^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{4} \cdot 5^{4} \cdot 23^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{12} \cdot 3^{4} \cdot 5^{4} \cdot 23^{4}\)
Sign: $1$
Analytic conductor: \(3.59968\)
Root analytic conductor: \(1.17363\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{12} \cdot 3^{4} \cdot 5^{4} \cdot 23^{4} ,\ ( \ : 0, 0, 0, 0 ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(11.36352353\)
\(L(\frac12)\) \(\approx\) \(11.36352353\)
\(L(1)\) 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 \( 1 \)
3$C_1$ \( ( 1 - T )^{4} \)
5$C_2^2$ \( 1 + T^{4} \)
23$C_2^2$ \( 1 + T^{4} \)
good7$C_2^3$ \( 1 - T^{4} + T^{8} \)
11$C_2$ \( ( 1 + T^{2} )^{4} \)
13$C_1$$\times$$C_2$ \( ( 1 - T )^{4}( 1 + T^{2} )^{2} \)
17$C_2^3$ \( 1 - T^{4} + T^{8} \)
19$C_2$ \( ( 1 + T^{2} )^{4} \)
29$C_2$ \( ( 1 + T + T^{2} )^{4} \)
31$C_2$ \( ( 1 + T + T^{2} )^{4} \)
37$C_2^3$ \( 1 - T^{4} + T^{8} \)
41$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
43$C_2^2$ \( ( 1 + T^{4} )^{2} \)
47$C_2^2$ \( ( 1 + T^{4} )^{2} \)
53$C_2^3$ \( 1 - T^{4} + T^{8} \)
59$C_2$ \( ( 1 - T + T^{2} )^{4} \)
61$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
67$C_2^3$ \( 1 - T^{4} + T^{8} \)
71$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
73$C_2^2$ \( ( 1 + T^{4} )^{2} \)
79$C_2^2$ \( ( 1 + T^{4} )^{2} \)
83$C_2^3$ \( 1 - T^{4} + T^{8} \)
89$C_2^2$ \( ( 1 + T^{4} )^{2} \)
97$C_2^2$ \( ( 1 + T^{4} )^{2} \)
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

−6.60418449535719205467423978239, −6.30800836710070171479535163755, −6.02971156557631318241108877641, −5.84488189149506771085838082633, −5.79279385219315183569770941731, −5.31215820555109293747284713240, −5.11042990642669460957400026336, −5.03968045548116239167462363518, −4.78083806288620263499399615810, −4.07673682085867526693422109651, −3.98164640975534348231637770703, −3.91837664815999178445824907671, −3.87218810985950202495635614610, −3.71060429016024908511989568801, −3.47713980513251299279468740598, −3.36296453139451773464353957715, −3.30711193704422804543908101268, −2.53658278465525417255501957901, −2.44095352682031156556609700009, −2.28687836999591453129113285420, −2.19492203667401294203760853353, −1.52041477015295031960808740563, −1.49355093912207211627991245028, −1.38055526986669349863331196469, −1.17821129963383442425022917312, 1.17821129963383442425022917312, 1.38055526986669349863331196469, 1.49355093912207211627991245028, 1.52041477015295031960808740563, 2.19492203667401294203760853353, 2.28687836999591453129113285420, 2.44095352682031156556609700009, 2.53658278465525417255501957901, 3.30711193704422804543908101268, 3.36296453139451773464353957715, 3.47713980513251299279468740598, 3.71060429016024908511989568801, 3.87218810985950202495635614610, 3.91837664815999178445824907671, 3.98164640975534348231637770703, 4.07673682085867526693422109651, 4.78083806288620263499399615810, 5.03968045548116239167462363518, 5.11042990642669460957400026336, 5.31215820555109293747284713240, 5.79279385219315183569770941731, 5.84488189149506771085838082633, 6.02971156557631318241108877641, 6.30800836710070171479535163755, 6.60418449535719205467423978239

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