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Properties

Label 8-3332e4-1.1-c0e4-0-7
Degree $8$
Conductor $1.233\times 10^{14}$
Sign $1$
Analytic cond. $7.64624$
Root an. cond. $1.28952$
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  + 4·5-s − 16-s + 10·25-s − 4·37-s − 4·53-s + 4·73-s − 4·80-s + 4·97-s + 4·109-s + 20·125-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 − 16·185-s + 191-s + 193-s + 197-s + ⋯
L(s)  = 1  + 4·5-s − 16-s + 10·25-s − 4·37-s − 4·53-s + 4·73-s − 4·80-s + 4·97-s + 4·109-s + 20·125-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 − 16·185-s + 191-s + 193-s + 197-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 7^{8} \cdot 17^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 7^{8} \cdot 17^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{8} \cdot 7^{8} \cdot 17^{4}\)
Sign: $1$
Analytic conductor: \(7.64624\)
Root analytic conductor: \(1.28952\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{8} \cdot 7^{8} \cdot 17^{4} ,\ ( \ : 0, 0, 0, 0 ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(4.161278239\)
\(L(\frac12)\) \(\approx\) \(4.161278239\)
\(L(1)\) 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$C_2^2$ \( 1 + T^{4} \)
7 \( 1 \)
17$C_2^2$ \( 1 + T^{4} \)
good3$C_4\times C_2$ \( 1 + T^{8} \)
5$C_1$$\times$$C_2^2$ \( ( 1 - T )^{4}( 1 + T^{4} ) \)
11$C_4\times C_2$ \( 1 + T^{8} \)
13$C_2$ \( ( 1 + T^{2} )^{4} \)
19$C_2^2$ \( ( 1 + T^{4} )^{2} \)
23$C_4\times C_2$ \( 1 + T^{8} \)
29$C_2$$\times$$C_2^2$ \( ( 1 + T^{2} )^{2}( 1 + T^{4} ) \)
31$C_4\times C_2$ \( 1 + T^{8} \)
37$C_1$$\times$$C_2^2$ \( ( 1 + T )^{4}( 1 + T^{4} ) \)
41$C_2$$\times$$C_2^2$ \( ( 1 + T^{2} )^{2}( 1 + T^{4} ) \)
43$C_2^2$ \( ( 1 + T^{4} )^{2} \)
47$C_2$ \( ( 1 + T^{2} )^{4} \)
53$C_1$$\times$$C_2$ \( ( 1 + T )^{4}( 1 + T^{2} )^{2} \)
59$C_2^2$ \( ( 1 + T^{4} )^{2} \)
61$C_2$$\times$$C_2^2$ \( ( 1 + T^{2} )^{2}( 1 + T^{4} ) \)
67$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
71$C_4\times C_2$ \( 1 + T^{8} \)
73$C_1$$\times$$C_2^2$ \( ( 1 - T )^{4}( 1 + T^{4} ) \)
79$C_4\times C_2$ \( 1 + T^{8} \)
83$C_2^2$ \( ( 1 + T^{4} )^{2} \)
89$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
97$C_1$$\times$$C_2^2$ \( ( 1 - T )^{4}( 1 + T^{4} ) \)
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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

−6.28704985212214390620944585903, −6.05519717658780310464426052651, −5.92205775272567661619324465982, −5.73311025379446444825846240177, −5.65451841112312950090643056823, −5.12505264609351885513588038136, −5.06227151101061327695043580417, −5.00870124223874830053303421204, −4.83374003956676857481882806514, −4.76587169000204867696532546397, −4.51237447014751953929728692299, −4.02956970599213676033029826720, −3.62944143387515088405518473871, −3.50938111775423619050128459432, −3.28716366349171627962390798530, −3.07932074125100982915405324204, −2.95616701517561944424574615945, −2.39861073835203771922515047313, −2.25033216888217488763691248948, −2.15841504260812958125026092373, −1.84739879112385287718000702336, −1.81031732133520589114588799588, −1.46124516365071390515862013025, −1.15154846705967676799574009531, −0.70358521393780651185275790086, 0.70358521393780651185275790086, 1.15154846705967676799574009531, 1.46124516365071390515862013025, 1.81031732133520589114588799588, 1.84739879112385287718000702336, 2.15841504260812958125026092373, 2.25033216888217488763691248948, 2.39861073835203771922515047313, 2.95616701517561944424574615945, 3.07932074125100982915405324204, 3.28716366349171627962390798530, 3.50938111775423619050128459432, 3.62944143387515088405518473871, 4.02956970599213676033029826720, 4.51237447014751953929728692299, 4.76587169000204867696532546397, 4.83374003956676857481882806514, 5.00870124223874830053303421204, 5.06227151101061327695043580417, 5.12505264609351885513588038136, 5.65451841112312950090643056823, 5.73311025379446444825846240177, 5.92205775272567661619324465982, 6.05519717658780310464426052651, 6.28704985212214390620944585903

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