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Properties

Label 8-570e4-1.1-c1e4-0-5
Degree $8$
Conductor $105560010000$
Sign $1$
Analytic cond. $429.148$
Root an. cond. $2.13341$
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  − 4·2-s − 4·3-s + 10·4-s + 16·6-s + 8·7-s − 20·8-s + 6·9-s − 40·12-s − 32·14-s + 35·16-s − 24·18-s − 32·21-s + 80·24-s − 2·25-s + 4·27-s + 80·28-s + 8·29-s − 56·32-s + 60·36-s + 16·41-s + 128·42-s + 32·43-s − 140·48-s + 16·49-s + 8·50-s + 24·53-s − 16·54-s + ⋯
L(s)  = 1  − 2.82·2-s − 2.30·3-s + 5·4-s + 6.53·6-s + 3.02·7-s − 7.07·8-s + 2·9-s − 11.5·12-s − 8.55·14-s + 35/4·16-s − 5.65·18-s − 6.98·21-s + 16.3·24-s − 2/5·25-s + 0.769·27-s + 15.1·28-s + 1.48·29-s − 9.89·32-s + 10·36-s + 2.49·41-s + 19.7·42-s + 4.87·43-s − 20.2·48-s + 16/7·49-s + 1.13·50-s + 3.29·53-s − 2.17·54-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 19^{4}\)
Sign: $1$
Analytic conductor: \(429.148\)
Root analytic conductor: \(2.13341\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{570} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 19^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.5165325773\)
\(L(\frac12)\) \(\approx\) \(0.5165325773\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 + T )^{4} \)
3$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
5$C_2$ \( ( 1 + T^{2} )^{2} \)
19$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \)
good7$D_{4}$ \( ( 1 - 4 T + 16 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \)
11$D_4\times C_2$ \( 1 - 8 T^{2} + 130 T^{4} - 8 p^{2} T^{6} + p^{4} T^{8} \)
13$D_4\times C_2$ \( 1 - 8 T^{2} + 66 T^{4} - 8 p^{2} T^{6} + p^{4} T^{8} \)
17$C_2^2$ \( ( 1 - 26 T^{2} + p^{2} T^{4} )^{2} \)
23$C_2^2$ \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \)
29$D_{4}$ \( ( 1 - 4 T + 12 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \)
31$D_4\times C_2$ \( 1 - 76 T^{2} + 2854 T^{4} - 76 p^{2} T^{6} + p^{4} T^{8} \)
37$D_4\times C_2$ \( 1 - 104 T^{2} + 5154 T^{4} - 104 p^{2} T^{6} + p^{4} T^{8} \)
41$D_{4}$ \( ( 1 - 8 T + 48 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \)
43$D_{4}$ \( ( 1 - 16 T + 148 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{2} \)
47$D_4\times C_2$ \( 1 - 100 T^{2} + 5766 T^{4} - 100 p^{2} T^{6} + p^{4} T^{8} \)
53$D_{4}$ \( ( 1 - 12 T + 134 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \)
59$D_{4}$ \( ( 1 - 20 T + 210 T^{2} - 20 p T^{3} + p^{2} T^{4} )^{2} \)
61$D_{4}$ \( ( 1 - 8 T + 66 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \)
67$C_2$ \( ( 1 - p T^{2} )^{4} \)
71$C_2^2$ \( ( 1 + 134 T^{2} + p^{2} T^{4} )^{2} \)
73$D_{4}$ \( ( 1 + 12 T + 110 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \)
79$D_4\times C_2$ \( 1 - 52 T^{2} + 11110 T^{4} - 52 p^{2} T^{6} + p^{4} T^{8} \)
83$D_4\times C_2$ \( 1 - 116 T^{2} + 13942 T^{4} - 116 p^{2} T^{6} + p^{4} T^{8} \)
89$D_{4}$ \( ( 1 + 24 T + 320 T^{2} + 24 p T^{3} + p^{2} T^{4} )^{2} \)
97$D_4\times C_2$ \( 1 - 256 T^{2} + 34690 T^{4} - 256 p^{2} T^{6} + p^{4} T^{8} \)
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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

−8.002491175347816181301896050313, −7.28563600338662704195527734285, −7.27667189790026424773862825380, −7.21890197136782646176877375883, −7.14946053955124206373481033098, −6.52713972028414602226997519436, −6.51095389077325742398044562978, −5.98540694012270788031111082900, −5.88946852933604129609279160345, −5.50175612034016594296986682554, −5.47284407434638017747440805165, −5.37646944716733884808360809484, −5.03887163466081568366153761917, −4.33966130192215834362048942823, −4.30541185027600792492673443088, −4.22122000047918702823790887336, −3.81374848090935712154043440781, −2.88520835534090871236014342005, −2.58225121220856977310922473630, −2.33839329309589272615354702397, −2.31484195027995799611342602018, −1.35657485493794321942914727266, −1.30297792593095552895869422482, −0.804001058421859420290469090962, −0.67745181899837192652731917851, 0.67745181899837192652731917851, 0.804001058421859420290469090962, 1.30297792593095552895869422482, 1.35657485493794321942914727266, 2.31484195027995799611342602018, 2.33839329309589272615354702397, 2.58225121220856977310922473630, 2.88520835534090871236014342005, 3.81374848090935712154043440781, 4.22122000047918702823790887336, 4.30541185027600792492673443088, 4.33966130192215834362048942823, 5.03887163466081568366153761917, 5.37646944716733884808360809484, 5.47284407434638017747440805165, 5.50175612034016594296986682554, 5.88946852933604129609279160345, 5.98540694012270788031111082900, 6.51095389077325742398044562978, 6.52713972028414602226997519436, 7.14946053955124206373481033098, 7.21890197136782646176877375883, 7.27667189790026424773862825380, 7.28563600338662704195527734285, 8.002491175347816181301896050313

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