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Properties

Label 8-12e8-1.1-c14e4-0-0
Degree $8$
Conductor $429981696$
Sign $1$
Analytic cond. $1.02739\times 10^{9}$
Root an. cond. $13.3803$
Motivic weight $14$
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  − 104·13-s − 1.22e10·25-s − 1.51e8·37-s + 1.35e12·49-s − 1.04e11·61-s − 1.01e12·73-s − 3.31e13·97-s − 1.31e14·109-s + 5.39e14·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 1.57e16·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯
L(s)  = 1  − 1.65e − 6·13-s − 2.00·25-s − 0.00159·37-s + 1.99·49-s − 0.0331·61-s − 0.0917·73-s − 0.410·97-s − 0.720·109-s + 1.42·121-s − 3.99·169-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(15-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+7)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{16} \cdot 3^{8}\)
Sign: $1$
Analytic conductor: \(1.02739\times 10^{9}\)
Root analytic conductor: \(13.3803\)
Motivic weight: \(14\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{16} \cdot 3^{8} ,\ ( \ : 7, 7, 7, 7 ),\ 1 )\)

Particular Values

\(L(\frac{15}{2})\) \(\approx\) \(0.08591186786\)
\(L(\frac12)\) \(\approx\) \(0.08591186786\)
\(L(8)\) 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 \( 1 \)
good5$C_2^2$ \( ( 1 + 1220742442 p T^{2} + p^{28} T^{4} )^{2} \)
7$C_2^2$ \( ( 1 - 96669855374 p T^{2} + p^{28} T^{4} )^{2} \)
11$C_2^2$ \( ( 1 - 24536943980662 p T^{2} + p^{28} T^{4} )^{2} \)
13$C_2$ \( ( 1 + 2 p T + p^{14} T^{2} )^{4} \)
17$C_2^2$ \( ( 1 + 336297440338554818 T^{2} + p^{28} T^{4} )^{2} \)
19$C_2^2$ \( ( 1 - 1450400026145747762 T^{2} + p^{28} T^{4} )^{2} \)
23$C_2^2$ \( ( 1 + 81758550235601182 T^{2} + p^{28} T^{4} )^{2} \)
29$C_2^2$ \( ( 1 + \)\(48\!\cdots\!02\)\( T^{2} + p^{28} T^{4} )^{2} \)
31$C_2^2$ \( ( 1 + \)\(56\!\cdots\!78\)\( T^{2} + p^{28} T^{4} )^{2} \)
37$C_2$ \( ( 1 + 37870486 T + p^{14} T^{2} )^{4} \)
41$C_2^2$ \( ( 1 - \)\(35\!\cdots\!98\)\( p T^{2} + p^{28} T^{4} )^{2} \)
43$C_2^2$ \( ( 1 - \)\(11\!\cdots\!78\)\( T^{2} + p^{28} T^{4} )^{2} \)
47$C_2^2$ \( ( 1 - \)\(47\!\cdots\!38\)\( T^{2} + p^{28} T^{4} )^{2} \)
53$C_2^2$ \( ( 1 - \)\(22\!\cdots\!22\)\( T^{2} + p^{28} T^{4} )^{2} \)
59$C_2^2$ \( ( 1 - \)\(10\!\cdots\!22\)\( T^{2} + p^{28} T^{4} )^{2} \)
61$C_2$ \( ( 1 + 26037385798 T + p^{14} T^{2} )^{4} \)
67$C_2^2$ \( ( 1 - \)\(63\!\cdots\!78\)\( T^{2} + p^{28} T^{4} )^{2} \)
71$C_2^2$ \( ( 1 - \)\(15\!\cdots\!62\)\( T^{2} + p^{28} T^{4} )^{2} \)
73$C_2$ \( ( 1 + 253367281586 T + p^{14} T^{2} )^{4} \)
79$C_2^2$ \( ( 1 - \)\(12\!\cdots\!42\)\( T^{2} + p^{28} T^{4} )^{2} \)
83$C_2^2$ \( ( 1 - \)\(30\!\cdots\!58\)\( T^{2} + p^{28} T^{4} )^{2} \)
89$C_2^2$ \( ( 1 + \)\(20\!\cdots\!22\)\( T^{2} + p^{28} T^{4} )^{2} \)
97$C_2$ \( ( 1 + 8291863420034 T + p^{14} T^{2} )^{4} \)
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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

−7.06537635271793322503410451332, −6.97569922578854759699167924868, −6.60823950713671954513891005605, −5.90613100738878254877772129426, −5.85295758193823656231523345142, −5.83527838067230568498120069961, −5.75433365448748740577073415004, −4.99827809707782278544859888434, −4.73981910991066871214030095636, −4.68370245413169664679035787165, −4.41543198554467846520115350392, −3.76655037104582693148901348239, −3.64873417753277343418620631013, −3.60981479262649058991446989222, −3.36577670602568411203111510862, −2.55985261294107115676558074290, −2.43836783324814589254442640164, −2.37245055293656652518360131889, −2.18615770895105740459749288218, −1.39067369437524491359105923465, −1.37587391272590344574053502899, −1.24392875917489659325599963346, −0.77849659674341722564677611541, −0.34473638563387814398667938176, −0.03769667398046693854305960947, 0.03769667398046693854305960947, 0.34473638563387814398667938176, 0.77849659674341722564677611541, 1.24392875917489659325599963346, 1.37587391272590344574053502899, 1.39067369437524491359105923465, 2.18615770895105740459749288218, 2.37245055293656652518360131889, 2.43836783324814589254442640164, 2.55985261294107115676558074290, 3.36577670602568411203111510862, 3.60981479262649058991446989222, 3.64873417753277343418620631013, 3.76655037104582693148901348239, 4.41543198554467846520115350392, 4.68370245413169664679035787165, 4.73981910991066871214030095636, 4.99827809707782278544859888434, 5.75433365448748740577073415004, 5.83527838067230568498120069961, 5.85295758193823656231523345142, 5.90613100738878254877772129426, 6.60823950713671954513891005605, 6.97569922578854759699167924868, 7.06537635271793322503410451332

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