Properties

Label 8-799e4-1.1-c0e4-0-1
Degree $8$
Conductor $407555836801$
Sign $1$
Analytic cond. $0.0252822$
Root an. cond. $0.631468$
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  + 4·9-s − 2·16-s − 4·17-s − 4·47-s + 4·49-s + 10·81-s + 127-s + 131-s + 137-s + 139-s − 8·144-s + 149-s + 151-s − 16·153-s + 157-s + 163-s + 167-s + 4·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + ⋯
L(s)  = 1  + 4·9-s − 2·16-s − 4·17-s − 4·47-s + 4·49-s + 10·81-s + 127-s + 131-s + 137-s + 139-s − 8·144-s + 149-s + 151-s − 16·153-s + 157-s + 163-s + 167-s + 4·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(17^{4} \cdot 47^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(17^{4} \cdot 47^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(17^{4} \cdot 47^{4}\)
Sign: $1$
Analytic conductor: \(0.0252822\)
Root analytic conductor: \(0.631468\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 17^{4} \cdot 47^{4} ,\ ( \ : 0, 0, 0, 0 ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8754587729\)
\(L(\frac12)\) \(\approx\) \(0.8754587729\)
\(L(1)\) 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)$
bad17$C_1$ \( ( 1 + T )^{4} \)
47$C_1$ \( ( 1 + T )^{4} \)
good2$C_2^2$ \( ( 1 + T^{4} )^{2} \)
3$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
5$C_4\times C_2$ \( 1 + T^{8} \)
7$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
11$C_4\times C_2$ \( 1 + T^{8} \)
13$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
19$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
23$C_4\times C_2$ \( 1 + T^{8} \)
29$C_4\times C_2$ \( 1 + T^{8} \)
31$C_4\times C_2$ \( 1 + T^{8} \)
37$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
41$C_4\times C_2$ \( 1 + T^{8} \)
43$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
53$C_2$ \( ( 1 + T^{2} )^{4} \)
59$C_2^2$ \( ( 1 + T^{4} )^{2} \)
61$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
67$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
71$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
73$C_4\times C_2$ \( 1 + T^{8} \)
79$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
83$C_2$ \( ( 1 + T^{2} )^{4} \)
89$C_2$ \( ( 1 + T^{2} )^{4} \)
97$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
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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

−7.49056534367430194780469798964, −7.25385161510358371747494107528, −6.95202882038181500529883090318, −6.86812145389592239968068311951, −6.82861976462304510514837837719, −6.60829086478723382339647557229, −6.49664325122377342082172518890, −6.01721691391608768666327825089, −5.96528377826056832212391592486, −5.36033789114543979090999479686, −4.89421399152344674744751737189, −4.84182717733109462279146146831, −4.81802217054357925804148446929, −4.44949415584839133562224720574, −4.16927173793250938157856854686, −4.09745807698964468747766934923, −3.95050209520928067550872210290, −3.64542438790764153491915610683, −2.99699198857455842145849223882, −2.73541741301945994096599588112, −2.14037838691091422830575316899, −2.11678113221751284338357849876, −1.87223478768312454428650563243, −1.58425295725740731501318166152, −0.889799171181315019954404780631, 0.889799171181315019954404780631, 1.58425295725740731501318166152, 1.87223478768312454428650563243, 2.11678113221751284338357849876, 2.14037838691091422830575316899, 2.73541741301945994096599588112, 2.99699198857455842145849223882, 3.64542438790764153491915610683, 3.95050209520928067550872210290, 4.09745807698964468747766934923, 4.16927173793250938157856854686, 4.44949415584839133562224720574, 4.81802217054357925804148446929, 4.84182717733109462279146146831, 4.89421399152344674744751737189, 5.36033789114543979090999479686, 5.96528377826056832212391592486, 6.01721691391608768666327825089, 6.49664325122377342082172518890, 6.60829086478723382339647557229, 6.82861976462304510514837837719, 6.86812145389592239968068311951, 6.95202882038181500529883090318, 7.25385161510358371747494107528, 7.49056534367430194780469798964

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