Properties

Label 8-300e4-1.1-c1e4-0-4
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  − 2·3-s + 4·7-s + 2·9-s + 12·13-s − 8·21-s − 6·27-s + 16·31-s + 12·37-s − 24·39-s + 12·43-s + 8·49-s − 24·61-s + 8·63-s + 4·67-s − 4·73-s + 11·81-s + 48·91-s − 32·93-s − 36·97-s − 4·103-s − 24·111-s + 24·117-s + 4·121-s + 127-s − 24·129-s + 131-s + 137-s + ⋯
L(s)  = 1  − 1.15·3-s + 1.51·7-s + 2/3·9-s + 3.32·13-s − 1.74·21-s − 1.15·27-s + 2.87·31-s + 1.97·37-s − 3.84·39-s + 1.82·43-s + 8/7·49-s − 3.07·61-s + 1.00·63-s + 0.488·67-s − 0.468·73-s + 11/9·81-s + 5.03·91-s − 3.31·93-s − 3.65·97-s − 0.394·103-s − 2.27·111-s + 2.21·117-s + 4/11·121-s + 0.0887·127-s − 2.11·129-s + 0.0873·131-s + 0.0854·137-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

\(L(1)\) \(\approx\) \(2.166063009\)
\(L(\frac12)\) \(\approx\) \(2.166063009\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
5 \( 1 \)
good7$C_2^2$ \( ( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \)
11$C_2^2$ \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \)
13$C_2^2$ \( ( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \)
17$C_2^3$ \( 1 - 2 T^{4} + p^{4} T^{8} \)
19$C_2^2$ \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{2} \)
23$C_2^3$ \( 1 + 238 T^{4} + p^{4} T^{8} \)
29$C_2^2$ \( ( 1 + 38 T^{2} + p^{2} T^{4} )^{2} \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )^{4} \)
37$C_2^2$ \( ( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \)
41$C_2^2$ \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \)
43$C_2^2$ \( ( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \)
47$C_2^2$$\times$$C_2^2$ \( ( 1 - 14 T + 98 T^{2} - 14 p T^{3} + p^{2} T^{4} )( 1 + 14 T + 98 T^{2} + 14 p T^{3} + p^{2} T^{4} ) \)
53$C_2^3$ \( 1 + 3598 T^{4} + p^{4} T^{8} \)
59$C_2^2$ \( ( 1 + 38 T^{2} + p^{2} T^{4} )^{2} \)
61$C_2$ \( ( 1 + 6 T + p T^{2} )^{4} \)
67$C_2^2$ \( ( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \)
71$C_2^2$ \( ( 1 - 122 T^{2} + p^{2} T^{4} )^{2} \)
73$C_2^2$ \( ( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \)
79$C_2^2$ \( ( 1 - 122 T^{2} + p^{2} T^{4} )^{2} \)
83$C_2^2$$\times$$C_2^2$ \( ( 1 - 22 T + 242 T^{2} - 22 p T^{3} + p^{2} T^{4} )( 1 + 22 T + 242 T^{2} + 22 p T^{3} + p^{2} T^{4} ) \)
89$C_2^2$ \( ( 1 + 158 T^{2} + p^{2} T^{4} )^{2} \)
97$C_2^2$ \( ( 1 + 18 T + 162 T^{2} + 18 p T^{3} + p^{2} T^{4} )^{2} \)
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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

−8.536673609276060105715686709295, −8.361699605012605427135800660692, −7.928900008981773018156893847854, −7.84159630611829227813236471232, −7.75805406001279014910180815952, −7.24355413676920567416071509382, −6.88018142651264645346305697017, −6.64238890294253049764031788464, −6.41670149233872565732193823297, −5.93317061011583505420907459422, −5.82979013310418854009129207376, −5.76113911921327072589666841386, −5.75087394928268454706433974125, −4.78111411192609383689351198856, −4.69823729292359967523597413842, −4.68610918289573643177172169771, −4.15978367529965748139362840761, −3.88826997094273363745296248787, −3.69176098081617736417805070568, −2.99439973037375114555512246422, −2.81455434271814746271916508568, −2.22398376156770181511611763688, −1.58007197925693039768896021020, −1.16587846828609069376831781271, −0.975701264454769177292084140345, 0.975701264454769177292084140345, 1.16587846828609069376831781271, 1.58007197925693039768896021020, 2.22398376156770181511611763688, 2.81455434271814746271916508568, 2.99439973037375114555512246422, 3.69176098081617736417805070568, 3.88826997094273363745296248787, 4.15978367529965748139362840761, 4.68610918289573643177172169771, 4.69823729292359967523597413842, 4.78111411192609383689351198856, 5.75087394928268454706433974125, 5.76113911921327072589666841386, 5.82979013310418854009129207376, 5.93317061011583505420907459422, 6.41670149233872565732193823297, 6.64238890294253049764031788464, 6.88018142651264645346305697017, 7.24355413676920567416071509382, 7.75805406001279014910180815952, 7.84159630611829227813236471232, 7.928900008981773018156893847854, 8.361699605012605427135800660692, 8.536673609276060105715686709295

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