Properties

Label 8-12e12-1.1-c0e4-0-0
Degree $8$
Conductor $8.916\times 10^{12}$
Sign $1$
Analytic cond. $0.553099$
Root an. cond. $0.928646$
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  − 4·13-s − 4·31-s + 4·43-s + 2·49-s − 4·67-s + 4·97-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 8·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + ⋯
L(s)  = 1  − 4·13-s − 4·31-s + 4·43-s + 2·49-s − 4·67-s + 4·97-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 8·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{24} \cdot 3^{12}\)
Sign: $1$
Analytic conductor: \(0.553099\)
Root analytic conductor: \(0.928646\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1728} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{24} \cdot 3^{12} ,\ ( \ : 0, 0, 0, 0 ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5899894966\)
\(L(\frac12)\) \(\approx\) \(0.5899894966\)
\(L(1)\) 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 \( 1 \)
good5$C_2^3$ \( 1 - T^{4} + T^{8} \)
7$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
11$C_2^3$ \( 1 - T^{4} + T^{8} \)
13$C_1$$\times$$C_2$ \( ( 1 + T )^{4}( 1 + T^{2} )^{2} \)
17$C_2^2$ \( ( 1 + T^{4} )^{2} \)
19$C_2^2$ \( ( 1 + T^{4} )^{2} \)
23$C_2$ \( ( 1 + T^{2} )^{4} \)
29$C_2^2$ \( ( 1 + T^{4} )^{2} \)
31$C_2$ \( ( 1 + T + T^{2} )^{4} \)
37$C_2^2$ \( ( 1 + T^{4} )^{2} \)
41$C_2$ \( ( 1 + T^{2} )^{4} \)
43$C_1$$\times$$C_2$ \( ( 1 - T )^{4}( 1 + T^{2} )^{2} \)
47$C_2^2$ \( ( 1 + T^{4} )^{2} \)
53$C_2^3$ \( 1 - T^{4} + T^{8} \)
59$C_2^2$ \( ( 1 + T^{4} )^{2} \)
61$C_2^2$ \( ( 1 + T^{4} )^{2} \)
67$C_1$$\times$$C_2$ \( ( 1 + T )^{4}( 1 + T^{2} )^{2} \)
71$C_2^2$ \( ( 1 + T^{4} )^{2} \)
73$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
79$C_2$ \( ( 1 + T^{2} )^{4} \)
83$C_2^3$ \( 1 - T^{4} + T^{8} \)
89$C_2^2$ \( ( 1 + T^{4} )^{2} \)
97$C_2$ \( ( 1 - T + T^{2} )^{4} \)
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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

−7.06455436619895528217560852122, −6.86525923532066924452135072959, −6.30731665863474286880736414604, −6.17304769351559242434115416871, −5.79240394399931854780880881073, −5.78136607915405100082449493527, −5.61786269038940756551980183896, −5.37494337854309505315929996393, −5.16747775415746864305776754321, −4.86704113590979641538307619655, −4.52027588007608767883237291715, −4.50149724821063543432988127956, −4.39978722555397598192815622328, −4.06423382406103284418400831694, −3.67198967629141611279574612688, −3.51119577129881542102389610905, −3.13588063935268142939853697068, −2.96373279465201844086410963084, −2.57643658410204539008691649513, −2.38905442266244460134343487945, −2.19137392873903803239592831973, −1.86249989552766588058839617478, −1.76854226414883281379709208327, −0.988546802046507834733748318804, −0.45533861842714877033138907550, 0.45533861842714877033138907550, 0.988546802046507834733748318804, 1.76854226414883281379709208327, 1.86249989552766588058839617478, 2.19137392873903803239592831973, 2.38905442266244460134343487945, 2.57643658410204539008691649513, 2.96373279465201844086410963084, 3.13588063935268142939853697068, 3.51119577129881542102389610905, 3.67198967629141611279574612688, 4.06423382406103284418400831694, 4.39978722555397598192815622328, 4.50149724821063543432988127956, 4.52027588007608767883237291715, 4.86704113590979641538307619655, 5.16747775415746864305776754321, 5.37494337854309505315929996393, 5.61786269038940756551980183896, 5.78136607915405100082449493527, 5.79240394399931854780880881073, 6.17304769351559242434115416871, 6.30731665863474286880736414604, 6.86525923532066924452135072959, 7.06455436619895528217560852122

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