Properties

Label 8-12e8-1.1-c1e4-0-4
Degree $8$
Conductor $429981696$
Sign $1$
Analytic cond. $1.74806$
Root an. cond. $1.07230$
Motivic weight $1$
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  + 6·5-s + 6·9-s − 2·13-s + 11·25-s − 30·29-s − 16·37-s + 18·41-s + 36·45-s − 5·49-s + 14·61-s − 12·65-s + 16·73-s + 27·81-s − 2·97-s − 18·101-s + 16·109-s − 42·113-s − 12·117-s − 5·121-s − 6·125-s + 127-s + 131-s + 137-s + 139-s − 180·145-s + 149-s + 151-s + ⋯
L(s)  = 1  + 2.68·5-s + 2·9-s − 0.554·13-s + 11/5·25-s − 5.57·29-s − 2.63·37-s + 2.81·41-s + 5.36·45-s − 5/7·49-s + 1.79·61-s − 1.48·65-s + 1.87·73-s + 3·81-s − 0.203·97-s − 1.79·101-s + 1.53·109-s − 3.95·113-s − 1.10·117-s − 0.454·121-s − 0.536·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 14.9·145-s + 0.0819·149-s + 0.0813·151-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{16} \cdot 3^{8}\)
Sign: $1$
Analytic conductor: \(1.74806\)
Root analytic conductor: \(1.07230\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{16} \cdot 3^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(2.159368795\)
\(L(\frac12)\) \(\approx\) \(2.159368795\)
\(L(\frac{3}{2})\) 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 \( 1 \)
3$C_2$ \( ( 1 - p T^{2} )^{2} \)
good5$C_2^2$ \( ( 1 - 3 T + 8 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \)
7$C_2^3$ \( 1 + 5 T^{2} - 24 T^{4} + 5 p^{2} T^{6} + p^{4} T^{8} \)
11$C_2^3$ \( 1 + 5 T^{2} - 96 T^{4} + 5 p^{2} T^{6} + p^{4} T^{8} \)
13$C_2^2$ \( ( 1 + T - 12 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \)
17$C_2^2$ \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \)
19$C_2^2$ \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \)
23$C_2^3$ \( 1 - 19 T^{2} - 168 T^{4} - 19 p^{2} T^{6} + p^{4} T^{8} \)
29$C_2^2$ \( ( 1 + 15 T + 104 T^{2} + 15 p T^{3} + p^{2} T^{4} )^{2} \)
31$C_2^3$ \( 1 + 53 T^{2} + 1848 T^{4} + 53 p^{2} T^{6} + p^{4} T^{8} \)
37$C_2$ \( ( 1 + 4 T + p T^{2} )^{4} \)
41$C_2^2$ \( ( 1 - 9 T + 68 T^{2} - 9 p T^{3} + p^{2} T^{4} )^{2} \)
43$C_2^3$ \( 1 + 77 T^{2} + 4080 T^{4} + 77 p^{2} T^{6} + p^{4} T^{8} \)
47$C_2^3$ \( 1 - 67 T^{2} + 2280 T^{4} - 67 p^{2} T^{6} + p^{4} T^{8} \)
53$C_2^2$ \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \)
59$C_2^3$ \( 1 - 91 T^{2} + 4800 T^{4} - 91 p^{2} T^{6} + p^{4} T^{8} \)
61$C_2^2$ \( ( 1 - 7 T - 12 T^{2} - 7 p T^{3} + p^{2} T^{4} )^{2} \)
67$C_2^3$ \( 1 + 53 T^{2} - 1680 T^{4} + 53 p^{2} T^{6} + p^{4} T^{8} \)
71$C_2^2$ \( ( 1 + 34 T^{2} + p^{2} T^{4} )^{2} \)
73$C_2$ \( ( 1 - 4 T + p T^{2} )^{4} \)
79$C_2^3$ \( 1 - 67 T^{2} - 1752 T^{4} - 67 p^{2} T^{6} + p^{4} T^{8} \)
83$C_2^3$ \( 1 - 139 T^{2} + 12432 T^{4} - 139 p^{2} T^{6} + p^{4} T^{8} \)
89$C_2^2$ \( ( 1 - 166 T^{2} + p^{2} T^{4} )^{2} \)
97$C_2^2$ \( ( 1 + T - 96 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \)
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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

−9.694107273038700793747814262090, −9.324720843994945638111076555549, −9.247501654964926554511061083515, −9.076031765902543733084110168792, −8.930586393799292921939680753528, −8.008019579604134389172515308716, −7.86362587493212970321374400110, −7.71201406813773082747995185529, −7.40006653335706045424681025361, −6.87550645366045446450862709724, −6.77606345060919167803736925013, −6.65780360780015032693371357846, −6.01561271083877878439268909073, −5.69301998497479816839619117232, −5.54237733388065670401573669277, −5.29203905096167257874834818048, −5.17852515331783687498942647739, −4.33263855178206094586604876304, −4.14211526011788941498503256286, −3.69268206051162075448026450011, −3.48429164753583361322936142108, −2.46632656362164108024302273549, −2.13731651477474007062868200648, −1.77323529395990812719051008617, −1.59850401755550870420465702409, 1.59850401755550870420465702409, 1.77323529395990812719051008617, 2.13731651477474007062868200648, 2.46632656362164108024302273549, 3.48429164753583361322936142108, 3.69268206051162075448026450011, 4.14211526011788941498503256286, 4.33263855178206094586604876304, 5.17852515331783687498942647739, 5.29203905096167257874834818048, 5.54237733388065670401573669277, 5.69301998497479816839619117232, 6.01561271083877878439268909073, 6.65780360780015032693371357846, 6.77606345060919167803736925013, 6.87550645366045446450862709724, 7.40006653335706045424681025361, 7.71201406813773082747995185529, 7.86362587493212970321374400110, 8.008019579604134389172515308716, 8.930586393799292921939680753528, 9.076031765902543733084110168792, 9.247501654964926554511061083515, 9.324720843994945638111076555549, 9.694107273038700793747814262090

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