Properties

Degree $8$
Conductor $2.560\times 10^{14}$
Sign $1$
Motivic weight $0$
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 9-s − 2·13-s − 4·17-s − 4·29-s + 4·41-s + 2·49-s − 2·53-s − 2·61-s − 2·73-s + 2·97-s + 4·101-s − 2·109-s + 4·113-s − 2·117-s + 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 4·153-s + 157-s + 163-s + 167-s + 169-s + 173-s + ⋯
L(s)  = 1  + 9-s − 2·13-s − 4·17-s − 4·29-s + 4·41-s + 2·49-s − 2·53-s − 2·61-s − 2·73-s + 2·97-s + 4·101-s − 2·109-s + 4·113-s − 2·117-s + 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 4·153-s + 157-s + 163-s + 167-s + 169-s + 173-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 5^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 5^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{20} \cdot 5^{12}\)
Sign: $1$
Motivic weight: \(0\)
Character: induced by $\chi_{4000} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{20} \cdot 5^{12} ,\ ( \ : 0, 0, 0, 0 ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5720994699\)
\(L(\frac12)\) \(\approx\) \(0.5720994699\)
\(L(1)\) 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 \( 1 \)
good3$C_4\times C_2$ \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \)
7$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
11$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
13$C_4$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
17$C_2$ \( ( 1 + T + T^{2} )^{4} \)
19$C_4\times C_2$ \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \)
23$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
29$C_2$ \( ( 1 + T + T^{2} )^{4} \)
31$C_4\times C_2$ \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \)
37$C_2$ \( ( 1 + T^{2} )^{4} \)
41$C_2$ \( ( 1 - T + T^{2} )^{4} \)
43$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
47$C_4\times C_2$ \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \)
53$C_4$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
59$C_4\times C_2$ \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \)
61$C_4$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
67$C_4\times C_2$ \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \)
71$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
73$C_4$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
79$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
83$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
89$C_2$ \( ( 1 + T^{2} )^{4} \)
97$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \)
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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

−6.16307091778900805176193829291, −5.97569036803006198942424979041, −5.87338519582092453275473412395, −5.57281628010657758576924499425, −5.40817734288491161454754442675, −5.05116724938903643651891765014, −4.92229870832455759520692965289, −4.68061046561209791955590972037, −4.51178636250351241981734582257, −4.32464146740364387940538477509, −4.27921128706487759394645745623, −4.02583803413308960917667996309, −3.98671781572133616016195783297, −3.35866264594108745200205293175, −3.33921066949657398906188169899, −3.16281552439781541146264152255, −2.65755440860477085356212857283, −2.35624710858523781555697740030, −2.34021355019288633650008374236, −2.10026071645969598454327607370, −2.06816803820930480604423905075, −1.52140698720427526983637471325, −1.49857196349192244242743947173, −0.74316149484627271561463929145, −0.30685085800694714946030010550, 0.30685085800694714946030010550, 0.74316149484627271561463929145, 1.49857196349192244242743947173, 1.52140698720427526983637471325, 2.06816803820930480604423905075, 2.10026071645969598454327607370, 2.34021355019288633650008374236, 2.35624710858523781555697740030, 2.65755440860477085356212857283, 3.16281552439781541146264152255, 3.33921066949657398906188169899, 3.35866264594108745200205293175, 3.98671781572133616016195783297, 4.02583803413308960917667996309, 4.27921128706487759394645745623, 4.32464146740364387940538477509, 4.51178636250351241981734582257, 4.68061046561209791955590972037, 4.92229870832455759520692965289, 5.05116724938903643651891765014, 5.40817734288491161454754442675, 5.57281628010657758576924499425, 5.87338519582092453275473412395, 5.97569036803006198942424979041, 6.16307091778900805176193829291

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