Properties

Label 8-300e4-1.1-c1e4-0-3
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  + 4·4-s − 4·9-s + 12·16-s − 16·36-s + 8·49-s + 32·61-s + 32·64-s + 7·81-s + 64·109-s − 44·121-s + 127-s + 131-s + 137-s + 139-s − 48·144-s + 149-s + 151-s + 157-s + 163-s + 167-s − 52·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 32·196-s + ⋯
L(s)  = 1  + 2·4-s − 4/3·9-s + 3·16-s − 8/3·36-s + 8/7·49-s + 4.09·61-s + 4·64-s + 7/9·81-s + 6.13·109-s − 4·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 4·144-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 + 16/7·196-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

\(L(1)\) \(\approx\) \(3.109932181\)
\(L(\frac12)\) \(\approx\) \(3.109932181\)
\(L(\frac{3}{2})\) 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$C_2$ \( ( 1 - p T^{2} )^{2} \)
3$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \)
5 \( 1 \)
good7$C_2^2$ \( ( 1 - 4 T^{2} + p^{2} T^{4} )^{2} \)
11$C_2$ \( ( 1 + p T^{2} )^{4} \)
13$C_2$ \( ( 1 + p T^{2} )^{4} \)
17$C_2$ \( ( 1 - p T^{2} )^{4} \)
19$C_2$ \( ( 1 - p T^{2} )^{4} \)
23$C_2^2$ \( ( 1 + 44 T^{2} + p^{2} T^{4} )^{2} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )^{2}( 1 + 6 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - p T^{2} )^{4} \)
37$C_2$ \( ( 1 + p T^{2} )^{4} \)
41$C_2$ \( ( 1 - 12 T + p T^{2} )^{2}( 1 + 12 T + p T^{2} )^{2} \)
43$C_2^2$ \( ( 1 - 76 T^{2} + p^{2} T^{4} )^{2} \)
47$C_2^2$ \( ( 1 - 4 T^{2} + p^{2} T^{4} )^{2} \)
53$C_2$ \( ( 1 - p T^{2} )^{4} \)
59$C_2$ \( ( 1 + p T^{2} )^{4} \)
61$C_2$ \( ( 1 - 8 T + p T^{2} )^{4} \)
67$C_2^2$ \( ( 1 + 116 T^{2} + p^{2} T^{4} )^{2} \)
71$C_2$ \( ( 1 + p T^{2} )^{4} \)
73$C_2$ \( ( 1 + p T^{2} )^{4} \)
79$C_2$ \( ( 1 - p T^{2} )^{4} \)
83$C_2^2$ \( ( 1 - 76 T^{2} + p^{2} T^{4} )^{2} \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )^{2}( 1 + 6 T + p T^{2} )^{2} \)
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

−8.430185419055942607208072514309, −8.323886409467531876953720789446, −8.099889564467530799192740993572, −7.62346796558641939159658080491, −7.54662643931072457699598369477, −7.25271579290364933410954135621, −6.94745777030392306371897200553, −6.74966095109870690982018776324, −6.55760537413229617345144475204, −6.13738944215972820559193889449, −5.92609784288548712531003643413, −5.70347239274477912542262981235, −5.56387658201116029539267847275, −5.11244926398844520313607032892, −4.90793919715533224343542719492, −4.48114490032672114408415440880, −3.83745552114054671214763759206, −3.72870891484257363571015925605, −3.36592430985495431315241637102, −3.09350347636820903833734666410, −2.46962120675756535733012006642, −2.34879477507242529031403756799, −2.20948265216510993730510528330, −1.40846540252742740862468046415, −0.826896808199285001952706185328, 0.826896808199285001952706185328, 1.40846540252742740862468046415, 2.20948265216510993730510528330, 2.34879477507242529031403756799, 2.46962120675756535733012006642, 3.09350347636820903833734666410, 3.36592430985495431315241637102, 3.72870891484257363571015925605, 3.83745552114054671214763759206, 4.48114490032672114408415440880, 4.90793919715533224343542719492, 5.11244926398844520313607032892, 5.56387658201116029539267847275, 5.70347239274477912542262981235, 5.92609784288548712531003643413, 6.13738944215972820559193889449, 6.55760537413229617345144475204, 6.74966095109870690982018776324, 6.94745777030392306371897200553, 7.25271579290364933410954135621, 7.54662643931072457699598369477, 7.62346796558641939159658080491, 8.099889564467530799192740993572, 8.323886409467531876953720789446, 8.430185419055942607208072514309

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