Properties

Degree $8$
Conductor $1.549\times 10^{14}$
Sign $1$
Motivic weight $0$
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 4-s − 25-s − 6·31-s − 64-s − 2·79-s − 100-s − 121-s − 6·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-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-s − 25-s − 6·31-s − 64-s − 2·79-s − 100-s − 121-s − 6·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-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

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{12} \cdot 3^{8} \cdot 7^{8}\)
Sign: $1$
Motivic weight: \(0\)
Character: induced by $\chi_{3528} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{12} \cdot 3^{8} \cdot 7^{8} ,\ ( \ : 0, 0, 0, 0 ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2014906873\)
\(L(\frac12)\) \(\approx\) \(0.2014906873\)
\(L(1)\) 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$C_2^2$ \( 1 - T^{2} + T^{4} \)
3 \( 1 \)
7 \( 1 \)
good5$C_2$$\times$$C_2^2$ \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \)
11$C_2$$\times$$C_2^2$ \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \)
13$C_2$ \( ( 1 + T^{2} )^{4} \)
17$C_2$ \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \)
19$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
23$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
29$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
31$C_1$$\times$$C_2$ \( ( 1 + T )^{4}( 1 + T + T^{2} )^{2} \)
37$C_2$ \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \)
41$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
43$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
47$C_2$ \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \)
53$C_2$$\times$$C_2^2$ \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \)
59$C_2$$\times$$C_2^2$ \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \)
61$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
67$C_2$ \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \)
71$C_2$ \( ( 1 + T^{2} )^{4} \)
73$C_2$ \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \)
79$C_1$$\times$$C_2$ \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \)
83$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
89$C_2$ \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \)
97$C_2$ \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \)
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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

−6.22331809369899560425638436067, −6.01678690582957789126432851271, −5.75898325855555291322067703194, −5.73354281641219609440268816672, −5.48225692302743293197348066355, −5.40325792910807685579913033669, −5.03751937344618109071226814587, −4.98631094857944052906232788493, −4.71040650775833899404681191272, −4.31656233748741455226282786416, −4.19618467312770418752699434175, −3.85595229068464594085815814716, −3.84982242891029496866036603075, −3.52796638331368601987844280031, −3.36216327225643675048503068124, −3.28503536118662693449654919080, −2.80345752791468587936824401929, −2.49854503131194138573829266227, −2.29776339363999742878148785697, −2.27172396268634410937145333381, −1.81512168043010021678651177499, −1.65469086099711228253362739435, −1.30068320453542783647605866453, −1.26662938912428378822387550457, −0.14187585088491212629746735570, 0.14187585088491212629746735570, 1.26662938912428378822387550457, 1.30068320453542783647605866453, 1.65469086099711228253362739435, 1.81512168043010021678651177499, 2.27172396268634410937145333381, 2.29776339363999742878148785697, 2.49854503131194138573829266227, 2.80345752791468587936824401929, 3.28503536118662693449654919080, 3.36216327225643675048503068124, 3.52796638331368601987844280031, 3.84982242891029496866036603075, 3.85595229068464594085815814716, 4.19618467312770418752699434175, 4.31656233748741455226282786416, 4.71040650775833899404681191272, 4.98631094857944052906232788493, 5.03751937344618109071226814587, 5.40325792910807685579913033669, 5.48225692302743293197348066355, 5.73354281641219609440268816672, 5.75898325855555291322067703194, 6.01678690582957789126432851271, 6.22331809369899560425638436067

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