Properties

Label 8-2240e4-1.1-c0e4-0-0
Degree $8$
Conductor $2.518\times 10^{13}$
Sign $1$
Analytic cond. $1.56178$
Root an. cond. $1.05731$
Motivic weight $0$
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  − 2·25-s − 8·29-s − 2·81-s − 4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 4·169-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 + ⋯
L(s)  = 1  − 2·25-s − 8·29-s − 2·81-s − 4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 4·169-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 + ⋯

Functional equation

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

Invariants

Degree: \(8\)
Conductor: \(2^{24} \cdot 5^{4} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(1.56178\)
Root analytic conductor: \(1.05731\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{2240} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{24} \cdot 5^{4} \cdot 7^{4} ,\ ( \ : 0, 0, 0, 0 ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.06032388029\)
\(L(\frac12)\) \(\approx\) \(0.06032388029\)
\(L(1)\) 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$C_2$ \( ( 1 + T^{2} )^{2} \)
7$C_2^2$ \( 1 + T^{4} \)
good3$C_2^2$ \( ( 1 + T^{4} )^{2} \)
11$C_2$ \( ( 1 + T^{2} )^{4} \)
13$C_2$ \( ( 1 + T^{2} )^{4} \)
17$C_2$ \( ( 1 + T^{2} )^{4} \)
19$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
23$C_2^2$ \( ( 1 + T^{4} )^{2} \)
29$C_1$ \( ( 1 + T )^{8} \)
31$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
37$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
41$C_2$ \( ( 1 + T^{2} )^{4} \)
43$C_2^2$ \( ( 1 + T^{4} )^{2} \)
47$C_2^2$ \( ( 1 + T^{4} )^{2} \)
53$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
59$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
61$C_2$ \( ( 1 + T^{2} )^{4} \)
67$C_2^2$ \( ( 1 + T^{4} )^{2} \)
71$C_2$ \( ( 1 + T^{2} )^{4} \)
73$C_2$ \( ( 1 + T^{2} )^{4} \)
79$C_2$ \( ( 1 + T^{2} )^{4} \)
83$C_2^2$ \( ( 1 + T^{4} )^{2} \)
89$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
97$C_2$ \( ( 1 + T^{2} )^{4} \)
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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

−6.62931597850986804223674062900, −6.28011466778037113510506983890, −6.14641907453021981221506252895, −5.83920138823794587695111952648, −5.79656272267362009738392787127, −5.66869594877539609422565847783, −5.40423386802856022619193045079, −5.24609300655022092760602989252, −5.01204313542183171913037913994, −4.89356266618034015865649900991, −4.20407433506505993895151178118, −4.15552479623073398087283616812, −4.05178209043252604440803154498, −3.93054745137848606669445419162, −3.51556376065953967518053034108, −3.42050594435959414388165862574, −3.35818709159905837849051285624, −2.78734616764231358974668198617, −2.47430116350105595534480659430, −2.17493756776900941870296590072, −1.99226898323628789385560294789, −1.84924287473662587634657924794, −1.36494030425422505290753424538, −1.34666811078089520755033665187, −0.10584117177015804727069645033, 0.10584117177015804727069645033, 1.34666811078089520755033665187, 1.36494030425422505290753424538, 1.84924287473662587634657924794, 1.99226898323628789385560294789, 2.17493756776900941870296590072, 2.47430116350105595534480659430, 2.78734616764231358974668198617, 3.35818709159905837849051285624, 3.42050594435959414388165862574, 3.51556376065953967518053034108, 3.93054745137848606669445419162, 4.05178209043252604440803154498, 4.15552479623073398087283616812, 4.20407433506505993895151178118, 4.89356266618034015865649900991, 5.01204313542183171913037913994, 5.24609300655022092760602989252, 5.40423386802856022619193045079, 5.66869594877539609422565847783, 5.79656272267362009738392787127, 5.83920138823794587695111952648, 6.14641907453021981221506252895, 6.28011466778037113510506983890, 6.62931597850986804223674062900

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