Properties

Label 8-63e8-1.1-c0e4-0-0
Degree $8$
Conductor $2.482\times 10^{14}$
Sign $1$
Analytic cond. $15.3940$
Root an. cond. $1.40740$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·4-s + 16-s − 2·25-s − 2·37-s − 2·43-s − 2·64-s − 4·67-s − 4·79-s − 4·100-s + 4·109-s − 121-s + 127-s + 131-s + 137-s + 139-s − 4·148-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2·169-s − 4·172-s + 173-s + 179-s + 181-s + 191-s + ⋯
L(s)  = 1  + 2·4-s + 16-s − 2·25-s − 2·37-s − 2·43-s − 2·64-s − 4·67-s − 4·79-s − 4·100-s + 4·109-s − 121-s + 127-s + 131-s + 137-s + 139-s − 4·148-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2·169-s − 4·172-s + 173-s + 179-s + 181-s + 191-s + ⋯

Functional equation

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

Invariants

Degree: \(8\)
Conductor: \(3^{16} \cdot 7^{8}\)
Sign: $1$
Analytic conductor: \(15.3940\)
Root analytic conductor: \(1.40740\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{3969} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 3^{16} \cdot 7^{8} ,\ ( \ : 0, 0, 0, 0 ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3752494930\)
\(L(\frac12)\) \(\approx\) \(0.3752494930\)
\(L(1)\) 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)$
bad3 \( 1 \)
7 \( 1 \)
good2$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
5$C_2$ \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \)
11$C_2$$\times$$C_2^2$ \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \)
13$C_2$ \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \)
17$C_2$ \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \)
19$C_2$ \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \)
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_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
37$C_1$$\times$$C_2$ \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \)
41$C_2$ \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \)
43$C_1$$\times$$C_2$ \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \)
47$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
53$C_2$$\times$$C_2^2$ \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \)
59$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
61$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
67$C_2$ \( ( 1 + T + T^{2} )^{4} \)
71$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
73$C_2$ \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \)
79$C_2$ \( ( 1 + T + T^{2} )^{4} \)
83$C_2$ \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \)
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

−6.19315876370603071433018479095, −6.14687332226755995032182462644, −5.74103451688420627806200466782, −5.59932681794536637332640113825, −5.54566697592830877382588787753, −5.34950153356990217512012478059, −4.86303142459192649329745923612, −4.76157698582531359048443328679, −4.52058801246556881069772529681, −4.51473506307316178015408403540, −4.19565742368966711099103853509, −3.87145888329129816713328400606, −3.52816982412546769549423176406, −3.47279728305178614986613421675, −3.23081534851153694707841920922, −3.21796899667575957517255585707, −2.58958799035717831687024240807, −2.52256023423446477942133308056, −2.43775079389472226373642379069, −2.19591301989554076321134192568, −1.71919932899013022003142714630, −1.52747262569354730044111210592, −1.38517412982404466569216303836, −1.38415930238760546561510544364, −0.16490124962486815178949979230, 0.16490124962486815178949979230, 1.38415930238760546561510544364, 1.38517412982404466569216303836, 1.52747262569354730044111210592, 1.71919932899013022003142714630, 2.19591301989554076321134192568, 2.43775079389472226373642379069, 2.52256023423446477942133308056, 2.58958799035717831687024240807, 3.21796899667575957517255585707, 3.23081534851153694707841920922, 3.47279728305178614986613421675, 3.52816982412546769549423176406, 3.87145888329129816713328400606, 4.19565742368966711099103853509, 4.51473506307316178015408403540, 4.52058801246556881069772529681, 4.76157698582531359048443328679, 4.86303142459192649329745923612, 5.34950153356990217512012478059, 5.54566697592830877382588787753, 5.59932681794536637332640113825, 5.74103451688420627806200466782, 6.14687332226755995032182462644, 6.19315876370603071433018479095

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