Properties

Label 8-3200e4-1.1-c0e4-0-6
Degree $8$
Conductor $1.049\times 10^{14}$
Sign $1$
Analytic cond. $6.50471$
Root an. cond. $1.26372$
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  + 5-s + 9-s + 2·17-s − 5·37-s − 2·41-s + 45-s + 4·49-s + 5·53-s + 2·73-s + 2·85-s − 3·89-s + 2·97-s + 3·113-s + 121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 2·153-s + 157-s + 163-s + 167-s − 169-s + 173-s + 179-s + ⋯
L(s)  = 1  + 5-s + 9-s + 2·17-s − 5·37-s − 2·41-s + 45-s + 4·49-s + 5·53-s + 2·73-s + 2·85-s − 3·89-s + 2·97-s + 3·113-s + 121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 2·153-s + 157-s + 163-s + 167-s − 169-s + 173-s + 179-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{28} \cdot 5^{8}\)
Sign: $1$
Analytic conductor: \(6.50471\)
Root analytic conductor: \(1.26372\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{28} \cdot 5^{8} ,\ ( \ : 0, 0, 0, 0 ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.708673522\)
\(L(\frac12)\) \(\approx\) \(2.708673522\)
\(L(1)\) 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 \( 1 \)
5$C_4$ \( 1 - T + T^{2} - T^{3} + T^{4} \)
good3$C_4\times C_2$ \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \)
7$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
11$C_4\times C_2$ \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \)
13$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
17$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \)
19$C_4\times C_2$ \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \)
23$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
29$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
31$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
37$C_1$$\times$$C_4$ \( ( 1 + T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} ) \)
41$C_4$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
43$C_2$ \( ( 1 + T^{2} )^{4} \)
47$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
53$C_1$$\times$$C_4$ \( ( 1 - T )^{4}( 1 - T + T^{2} - T^{3} + T^{4} ) \)
59$C_4\times C_2$ \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \)
61$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
67$C_4\times C_2$ \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \)
71$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
73$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \)
79$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
83$C_4\times C_2$ \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \)
89$C_1$$\times$$C_4$ \( ( 1 + T )^{4}( 1 - T + T^{2} - T^{3} + T^{4} ) \)
97$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \)
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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

−6.46666624929956502456626490401, −5.85300221237560484308325077907, −5.80789596988122099964611496146, −5.70331246113306750785785986948, −5.49061368615348345455993588123, −5.45873455142173248204850709313, −5.26522115555883223079368709631, −4.95676560386892569382700941266, −4.85161044254300700107330884179, −4.46236147769698298385916384452, −4.24958058728688162805137260719, −3.96773002813336913466049378123, −3.84092321382740838585372965437, −3.61755813301617864176875750554, −3.37337639601317900676233575704, −3.28971244864316759972261879328, −3.06118125411316418869301878784, −2.46331467358475224460943264113, −2.35589691724804671962656676417, −2.14165531233013198401759591911, −1.97463176479611492721024713556, −1.56617847762841779861088936776, −1.39951522512334665133599809721, −0.954130413471926463443190414300, −0.72858396334323566518485710608, 0.72858396334323566518485710608, 0.954130413471926463443190414300, 1.39951522512334665133599809721, 1.56617847762841779861088936776, 1.97463176479611492721024713556, 2.14165531233013198401759591911, 2.35589691724804671962656676417, 2.46331467358475224460943264113, 3.06118125411316418869301878784, 3.28971244864316759972261879328, 3.37337639601317900676233575704, 3.61755813301617864176875750554, 3.84092321382740838585372965437, 3.96773002813336913466049378123, 4.24958058728688162805137260719, 4.46236147769698298385916384452, 4.85161044254300700107330884179, 4.95676560386892569382700941266, 5.26522115555883223079368709631, 5.45873455142173248204850709313, 5.49061368615348345455993588123, 5.70331246113306750785785986948, 5.80789596988122099964611496146, 5.85300221237560484308325077907, 6.46666624929956502456626490401

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