Properties

Label 8-825e4-1.1-c1e4-0-20
Degree $8$
Conductor $463250390625$
Sign $1$
Analytic cond. $1883.32$
Root an. cond. $2.56664$
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 + 8·4-s + 16·6-s + 8·7-s + 12·8-s + 6·9-s + 32·12-s + 8·13-s + 32·14-s + 15·16-s + 24·18-s + 32·21-s + 4·23-s + 48·24-s + 32·26-s − 4·27-s + 64·28-s − 8·29-s − 24·31-s + 16·32-s + 48·36-s + 12·37-s + 32·39-s + 128·42-s + 16·43-s + 16·46-s + ⋯
L(s)  = 1  + 2.82·2-s + 2.30·3-s + 4·4-s + 6.53·6-s + 3.02·7-s + 4.24·8-s + 2·9-s + 9.23·12-s + 2.21·13-s + 8.55·14-s + 15/4·16-s + 5.65·18-s + 6.98·21-s + 0.834·23-s + 9.79·24-s + 6.27·26-s − 0.769·27-s + 12.0·28-s − 1.48·29-s − 4.31·31-s + 2.82·32-s + 8·36-s + 1.97·37-s + 5.12·39-s + 19.7·42-s + 2.43·43-s + 2.35·46-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(3^{4} \cdot 5^{8} \cdot 11^{4}\)
Sign: $1$
Analytic conductor: \(1883.32\)
Root analytic conductor: \(2.56664\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{825} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 3^{4} \cdot 5^{8} \cdot 11^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(71.74551872\)
\(L(\frac12)\) \(\approx\) \(71.74551872\)
\(L(\frac{3}{2})\) 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)$
bad3$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
5 \( 1 \)
11$C_2$ \( ( 1 + T^{2} )^{2} \)
good2$D_4\times C_2$ \( 1 - p^{2} T + p^{3} T^{2} - 3 p^{2} T^{3} + 17 T^{4} - 3 p^{3} T^{5} + p^{5} T^{6} - p^{5} T^{7} + p^{4} T^{8} \)
7$C_4\times C_2$ \( 1 - 8 T + 32 T^{2} - 88 T^{3} + 226 T^{4} - 88 p T^{5} + 32 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
13$C_2^2$ \( ( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \)
17$C_2^3$ \( 1 - 254 T^{4} + p^{4} T^{8} \)
19$C_2^2$ \( ( 1 - 30 T^{2} + p^{2} T^{4} )^{2} \)
23$D_4\times C_2$ \( 1 - 4 T + 8 T^{2} + 44 T^{3} - 914 T^{4} + 44 p T^{5} + 8 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
29$D_{4}$ \( ( 1 + 4 T + 54 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \)
31$D_{4}$ \( ( 1 + 12 T + 90 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \)
37$D_4\times C_2$ \( 1 - 12 T + 72 T^{2} - 468 T^{3} + 3038 T^{4} - 468 p T^{5} + 72 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \)
41$D_4\times C_2$ \( 1 - 92 T^{2} + 4966 T^{4} - 92 p^{2} T^{6} + p^{4} T^{8} \)
43$D_4\times C_2$ \( 1 - 16 T + 128 T^{2} - 624 T^{3} + 3026 T^{4} - 624 p T^{5} + 128 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \)
47$D_4\times C_2$ \( 1 - 12 T + 72 T^{2} - 348 T^{3} + 1358 T^{4} - 348 p T^{5} + 72 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \)
53$C_2^2$ \( ( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \)
59$C_2$ \( ( 1 + 4 T + p T^{2} )^{4} \)
61$D_{4}$ \( ( 1 - 4 T + 54 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \)
67$D_4\times C_2$ \( 1 - 20 T + 200 T^{2} - 2260 T^{3} + 23422 T^{4} - 2260 p T^{5} + 200 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} \)
71$D_4\times C_2$ \( 1 - 68 T^{2} + 870 T^{4} - 68 p^{2} T^{6} + p^{4} T^{8} \)
73$C_2^2$ \( ( 1 + p^{2} T^{4} )^{2} \)
79$D_4\times C_2$ \( 1 - 172 T^{2} + 17830 T^{4} - 172 p^{2} T^{6} + p^{4} T^{8} \)
83$D_4\times C_2$ \( 1 - 8 T + 32 T^{2} + 72 T^{3} - 8302 T^{4} + 72 p T^{5} + 32 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
89$D_{4}$ \( ( 1 + 8 T + 162 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \)
97$D_4\times C_2$ \( 1 + 20 T + 200 T^{2} + 1660 T^{3} + 13582 T^{4} + 1660 p T^{5} + 200 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} \)
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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

−7.34894779745338604898395644848, −7.18968710036931506022565682671, −7.13787540652739811304912687953, −6.56917759832328570109786603654, −6.13016205372987148376560684549, −5.98283576420688931687240586682, −5.74765079580935279076186878179, −5.66173687138756236383748585362, −5.39910480605603231706708573348, −5.07195815877535660133691697046, −5.00163778593237984102344560267, −4.53084510211319525980807390011, −4.46278373432925921509165297041, −4.01765575517228638209779848461, −3.95687024356980409401271330870, −3.71475029611568090445096209226, −3.68488520158486158013217938576, −3.31647290635706966181084980811, −2.95070071559490373775013961281, −2.49539654128689761425007508387, −2.21677995884008600009283489858, −2.15576498815130194013354878143, −1.79107502245595639152077496611, −1.26074146044966490430711807750, −1.21848365658770454487868104945, 1.21848365658770454487868104945, 1.26074146044966490430711807750, 1.79107502245595639152077496611, 2.15576498815130194013354878143, 2.21677995884008600009283489858, 2.49539654128689761425007508387, 2.95070071559490373775013961281, 3.31647290635706966181084980811, 3.68488520158486158013217938576, 3.71475029611568090445096209226, 3.95687024356980409401271330870, 4.01765575517228638209779848461, 4.46278373432925921509165297041, 4.53084510211319525980807390011, 5.00163778593237984102344560267, 5.07195815877535660133691697046, 5.39910480605603231706708573348, 5.66173687138756236383748585362, 5.74765079580935279076186878179, 5.98283576420688931687240586682, 6.13016205372987148376560684549, 6.56917759832328570109786603654, 7.13787540652739811304912687953, 7.18968710036931506022565682671, 7.34894779745338604898395644848

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