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Properties

Label 8-896e4-1.1-c0e4-0-0
Degree $8$
Conductor $644513529856$
Sign $1$
Analytic cond. $0.0399816$
Root an. cond. $0.668701$
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  + 4·23-s + 4·43-s − 4·53-s − 4·67-s − 4·107-s + 4·109-s − 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-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  + 4·23-s + 4·43-s − 4·53-s − 4·67-s − 4·107-s + 4·109-s − 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-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

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{28} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(0.0399816\)
Root analytic conductor: \(0.668701\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{28} \cdot 7^{4} ,\ ( \ : 0, 0, 0, 0 ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9405680311\)
\(L(\frac12)\) \(\approx\) \(0.9405680311\)
\(L(1)\) 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 \( 1 \)
7$C_2^2$ \( 1 + T^{4} \)
good3$C_4\times C_2$ \( 1 + T^{8} \)
5$C_4\times C_2$ \( 1 + T^{8} \)
11$C_2$$\times$$C_2^2$ \( ( 1 + T^{2} )^{2}( 1 + T^{4} ) \)
13$C_4\times C_2$ \( 1 + T^{8} \)
17$C_2$ \( ( 1 + T^{2} )^{4} \)
19$C_4\times C_2$ \( 1 + T^{8} \)
23$C_1$$\times$$C_2$ \( ( 1 - T )^{4}( 1 + T^{2} )^{2} \)
29$C_2$$\times$$C_2^2$ \( ( 1 + T^{2} )^{2}( 1 + T^{4} ) \)
31$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
37$C_2$$\times$$C_2^2$ \( ( 1 + T^{2} )^{2}( 1 + T^{4} ) \)
41$C_2^2$ \( ( 1 + T^{4} )^{2} \)
43$C_1$$\times$$C_2^2$ \( ( 1 - T )^{4}( 1 + T^{4} ) \)
47$C_2$ \( ( 1 + T^{2} )^{4} \)
53$C_1$$\times$$C_2^2$ \( ( 1 + T )^{4}( 1 + T^{4} ) \)
59$C_4\times C_2$ \( 1 + T^{8} \)
61$C_4\times C_2$ \( 1 + T^{8} \)
67$C_1$$\times$$C_2^2$ \( ( 1 + T )^{4}( 1 + T^{4} ) \)
71$C_2^2$ \( ( 1 + T^{4} )^{2} \)
73$C_2^2$ \( ( 1 + T^{4} )^{2} \)
79$C_2^2$ \( ( 1 + T^{4} )^{2} \)
83$C_4\times C_2$ \( 1 + T^{8} \)
89$C_2^2$ \( ( 1 + T^{4} )^{2} \)
97$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
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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

−7.57494709329354829425291154387, −7.26823706403039517249601387275, −7.05160542580676998808925786043, −6.77820815604410505503678097931, −6.66499596100028099353452090499, −6.28856204829249826704379628031, −6.03269040468560780819678370674, −5.99864746304311802393021972678, −5.74965680322947126483517689282, −5.23372999870034590157920533037, −5.05266164470524463309420157174, −5.03038666312973927882021152941, −4.68245350256690288926292139439, −4.51445647198220240170724483400, −4.04899665595233899823002489118, −3.98667709605783841911901767470, −3.67321395568854535224913615701, −3.01838524610478155287104693419, −2.90114223147121109838553647662, −2.90007727478073064167155416647, −2.75027351849633254106181210761, −2.00195672710065480054624641733, −1.68642899387736928874792203941, −1.17278319926453268689170926646, −1.02917740701153812391221961772, 1.02917740701153812391221961772, 1.17278319926453268689170926646, 1.68642899387736928874792203941, 2.00195672710065480054624641733, 2.75027351849633254106181210761, 2.90007727478073064167155416647, 2.90114223147121109838553647662, 3.01838524610478155287104693419, 3.67321395568854535224913615701, 3.98667709605783841911901767470, 4.04899665595233899823002489118, 4.51445647198220240170724483400, 4.68245350256690288926292139439, 5.03038666312973927882021152941, 5.05266164470524463309420157174, 5.23372999870034590157920533037, 5.74965680322947126483517689282, 5.99864746304311802393021972678, 6.03269040468560780819678370674, 6.28856204829249826704379628031, 6.66499596100028099353452090499, 6.77820815604410505503678097931, 7.05160542580676998808925786043, 7.26823706403039517249601387275, 7.57494709329354829425291154387

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