Properties

Label 8-300e4-1.1-c1e4-0-0
Degree $8$
Conductor $8100000000$
Sign $1$
Analytic cond. $32.9301$
Root an. cond. $1.54774$
Motivic weight $1$
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  − 44·31-s − 4·61-s − 9·81-s + 44·121-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 + 233-s + 239-s + 241-s + ⋯
L(s)  = 1  − 7.90·31-s − 0.512·61-s − 81-s + 4·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s + 0.0663·227-s + 0.0660·229-s + 0.0655·233-s + 0.0646·239-s + 0.0644·241-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{8} \cdot 3^{4} \cdot 5^{8}\)
Sign: $1$
Analytic conductor: \(32.9301\)
Root analytic conductor: \(1.54774\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{300} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{8} \cdot 3^{4} \cdot 5^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.9688981079\)
\(L(\frac12)\) \(\approx\) \(0.9688981079\)
\(L(\frac{3}{2})\) 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 \( 1 \)
3$C_2^2$ \( 1 + p^{2} T^{4} \)
5 \( 1 \)
good7$C_2^3$ \( 1 + 71 T^{4} + p^{4} T^{8} \)
11$C_2$ \( ( 1 - p T^{2} )^{4} \)
13$C_2^3$ \( 1 + 191 T^{4} + p^{4} T^{8} \)
17$C_2^2$ \( ( 1 + p^{2} T^{4} )^{2} \)
19$C_2^2$ \( ( 1 + 11 T^{2} + p^{2} T^{4} )^{2} \)
23$C_2^2$ \( ( 1 + p^{2} T^{4} )^{2} \)
29$C_2$ \( ( 1 + p T^{2} )^{4} \)
31$C_2$ \( ( 1 + 11 T + p T^{2} )^{4} \)
37$C_2^3$ \( 1 - 2062 T^{4} + p^{4} T^{8} \)
41$C_2$ \( ( 1 - p T^{2} )^{4} \)
43$C_2^3$ \( 1 + 3191 T^{4} + p^{4} T^{8} \)
47$C_2^2$ \( ( 1 + p^{2} T^{4} )^{2} \)
53$C_2^2$ \( ( 1 + p^{2} T^{4} )^{2} \)
59$C_2$ \( ( 1 + p T^{2} )^{4} \)
61$C_2$ \( ( 1 + T + p T^{2} )^{4} \)
67$C_2^3$ \( 1 - 8809 T^{4} + p^{4} T^{8} \)
71$C_2$ \( ( 1 - p T^{2} )^{4} \)
73$C_2^3$ \( 1 - 8542 T^{4} + p^{4} T^{8} \)
79$C_2^2$ \( ( 1 - 142 T^{2} + p^{2} T^{4} )^{2} \)
83$C_2^2$ \( ( 1 + p^{2} T^{4} )^{2} \)
89$C_2$ \( ( 1 + p T^{2} )^{4} \)
97$C_2^3$ \( 1 + 9071 T^{4} + p^{4} T^{8} \)
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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

−8.822555355197016147019460866317, −8.166704774242188692017644598861, −7.953120423714196314595464752976, −7.84148658730117904538530503067, −7.43842888431618224300548340974, −7.27063806508378292236530837756, −7.00924955764891364215982017054, −6.88498259536766895505327696159, −6.62190355994087854172886235568, −6.06431871183911066777028233098, −5.71177491798444827396276226959, −5.52482853605190452122363885586, −5.45229441999026992233723255918, −5.38283383068591988890539478739, −4.60886312772752862324915921247, −4.44020035648509998984483148507, −4.15313930557551341846262812536, −3.61653648692917148728047209142, −3.49490087929753217566214985753, −3.29679380227759339369699130625, −2.80015991950373825266521744282, −2.00105029732420890941231738876, −1.78628838782571669735172043363, −1.77973688557852886147883827335, −0.43487692495510207052947178929, 0.43487692495510207052947178929, 1.77973688557852886147883827335, 1.78628838782571669735172043363, 2.00105029732420890941231738876, 2.80015991950373825266521744282, 3.29679380227759339369699130625, 3.49490087929753217566214985753, 3.61653648692917148728047209142, 4.15313930557551341846262812536, 4.44020035648509998984483148507, 4.60886312772752862324915921247, 5.38283383068591988890539478739, 5.45229441999026992233723255918, 5.52482853605190452122363885586, 5.71177491798444827396276226959, 6.06431871183911066777028233098, 6.62190355994087854172886235568, 6.88498259536766895505327696159, 7.00924955764891364215982017054, 7.27063806508378292236530837756, 7.43842888431618224300548340974, 7.84148658730117904538530503067, 7.953120423714196314595464752976, 8.166704774242188692017644598861, 8.822555355197016147019460866317

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