Properties

Label 8-2240e4-1.1-c1e4-0-4
Degree $8$
Conductor $2.518\times 10^{13}$
Sign $1$
Analytic cond. $102352.$
Root an. cond. $4.22924$
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  − 8·9-s − 10·25-s − 24·29-s + 4·49-s + 30·81-s + 64·109-s + 44·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 52·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 80·225-s + ⋯
L(s)  = 1  − 8/3·9-s − 2·25-s − 4.45·29-s + 4/7·49-s + 10/3·81-s + 6.13·109-s + 4·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 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 + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s + 16/3·225-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{24} \cdot 5^{4} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(102352.\)
Root analytic conductor: \(4.22924\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{24} \cdot 5^{4} \cdot 7^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.4165087742\)
\(L(\frac12)\) \(\approx\) \(0.4165087742\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5$C_2$ \( ( 1 + p T^{2} )^{2} \)
7$C_2^2$ \( 1 - 4 T^{2} + p^{2} T^{4} \)
good3$C_2^2$ \( ( 1 + 4 T^{2} + p^{2} T^{4} )^{2} \)
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$C_2$ \( ( 1 + 6 T + p T^{2} )^{4} \)
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$C_2$ \( ( 1 - 8 T + p T^{2} )^{2}( 1 + 8 T + p T^{2} )^{2} \)
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

−6.18626744637382571576502939087, −6.16199721231586137252186261485, −6.15960520351229691892557538214, −5.74822505810721861857172927406, −5.55690075865132424130605400543, −5.45685524677552173695102718890, −5.40967421688039287204219122538, −4.91513420760546730349229050473, −4.75633790830764163800903661032, −4.69895265053037693066644683287, −4.06386846384179583269468146414, −4.02180998818222127083470969729, −3.81536893447775702771458449438, −3.45524386553575890633797234539, −3.43937411933734357360671393601, −3.13025275255520846070016035611, −3.03154078405596798249375947804, −2.35552498958665531634317790944, −2.25787638220547326247290733252, −2.13066588276179213650084271881, −2.05046882990868582126221356881, −1.43801247716543431557119253970, −1.15553645329425356869077312689, −0.42169270087196027488623536318, −0.18816949242081315049372663499, 0.18816949242081315049372663499, 0.42169270087196027488623536318, 1.15553645329425356869077312689, 1.43801247716543431557119253970, 2.05046882990868582126221356881, 2.13066588276179213650084271881, 2.25787638220547326247290733252, 2.35552498958665531634317790944, 3.03154078405596798249375947804, 3.13025275255520846070016035611, 3.43937411933734357360671393601, 3.45524386553575890633797234539, 3.81536893447775702771458449438, 4.02180998818222127083470969729, 4.06386846384179583269468146414, 4.69895265053037693066644683287, 4.75633790830764163800903661032, 4.91513420760546730349229050473, 5.40967421688039287204219122538, 5.45685524677552173695102718890, 5.55690075865132424130605400543, 5.74822505810721861857172927406, 6.15960520351229691892557538214, 6.16199721231586137252186261485, 6.18626744637382571576502939087

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