Properties

Label 8-300e4-1.1-c1e4-0-1
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  − 4-s + 6·9-s − 3·16-s − 6·36-s + 28·49-s − 8·61-s + 7·64-s + 27·81-s − 56·109-s − 44·121-s + 127-s + 131-s + 137-s + 139-s − 18·144-s + 149-s + 151-s + 157-s + 163-s + 167-s − 52·169-s + 173-s + 179-s + 181-s + 191-s + 193-s − 28·196-s + ⋯
L(s)  = 1  − 1/2·4-s + 2·9-s − 3/4·16-s − 36-s + 4·49-s − 1.02·61-s + 7/8·64-s + 3·81-s − 5.36·109-s − 4·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 3/2·144-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 4·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s − 2·196-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

\(L(1)\) \(\approx\) \(1.860502809\)
\(L(\frac12)\) \(\approx\) \(1.860502809\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 + T^{2} + p^{2} T^{4} \)
3$C_2$ \( ( 1 - p T^{2} )^{2} \)
5 \( 1 \)
good7$C_2$ \( ( 1 - p T^{2} )^{4} \)
11$C_2$ \( ( 1 + p T^{2} )^{4} \)
13$C_2$ \( ( 1 + p T^{2} )^{4} \)
17$C_2^2$ \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )^{2}( 1 + 4 T + p T^{2} )^{2} \)
23$C_2^2$ \( ( 1 + 34 T^{2} + p^{2} T^{4} )^{2} \)
29$C_2$ \( ( 1 - p T^{2} )^{4} \)
31$C_2$ \( ( 1 - 8 T + p T^{2} )^{2}( 1 + 8 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 + p T^{2} )^{4} \)
41$C_2$ \( ( 1 - p T^{2} )^{4} \)
43$C_2$ \( ( 1 - p T^{2} )^{4} \)
47$C_2^2$ \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \)
53$C_2^2$ \( ( 1 - 86 T^{2} + p^{2} T^{4} )^{2} \)
59$C_2$ \( ( 1 + p T^{2} )^{4} \)
61$C_2$ \( ( 1 + 2 T + p T^{2} )^{4} \)
67$C_2$ \( ( 1 - p T^{2} )^{4} \)
71$C_2$ \( ( 1 + p T^{2} )^{4} \)
73$C_2$ \( ( 1 + p T^{2} )^{4} \)
79$C_2$ \( ( 1 - 16 T + p T^{2} )^{2}( 1 + 16 T + p T^{2} )^{2} \)
83$C_2^2$ \( ( 1 + 154 T^{2} + p^{2} T^{4} )^{2} \)
89$C_2$ \( ( 1 - p T^{2} )^{4} \)
97$C_2$ \( ( 1 + p T^{2} )^{4} \)
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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

−8.834857352242505299862297906693, −8.072956056963245475517567963300, −8.042117752955831477681839279600, −7.83743197920996102622876190385, −7.53199024700368932420842637473, −7.14311312029625923470584536406, −7.09165322969349987606342131903, −6.79613763137732242710813556925, −6.56238621750688459368607035467, −6.12363996841608109918257343483, −6.11648504470573418558988894707, −5.37952273885941959794617987601, −5.24910132319504212874055274480, −5.21252859413873842000939297548, −4.70492407682606185113424540184, −4.22710936322728793186216207473, −4.09953679252022700648221542972, −4.00566856173883031188541835477, −3.74233123488470895105617752460, −3.03187788127953670040836800673, −2.58145881607263761644060066635, −2.45657223494028021055502165666, −1.69383047641562142343578541543, −1.43369041309139557867186498191, −0.68921781166120861241284768793, 0.68921781166120861241284768793, 1.43369041309139557867186498191, 1.69383047641562142343578541543, 2.45657223494028021055502165666, 2.58145881607263761644060066635, 3.03187788127953670040836800673, 3.74233123488470895105617752460, 4.00566856173883031188541835477, 4.09953679252022700648221542972, 4.22710936322728793186216207473, 4.70492407682606185113424540184, 5.21252859413873842000939297548, 5.24910132319504212874055274480, 5.37952273885941959794617987601, 6.11648504470573418558988894707, 6.12363996841608109918257343483, 6.56238621750688459368607035467, 6.79613763137732242710813556925, 7.09165322969349987606342131903, 7.14311312029625923470584536406, 7.53199024700368932420842637473, 7.83743197920996102622876190385, 8.042117752955831477681839279600, 8.072956056963245475517567963300, 8.834857352242505299862297906693

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