Properties

Label 8-825e4-1.1-c1e4-0-17
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 $4$

Origins

Origins of factors

Downloads

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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 + 8·9-s − 32·12-s − 8·13-s + 32·14-s + 15·16-s + 8·17-s − 32·18-s + 32·21-s − 8·23-s + 48·24-s + 32·26-s − 12·27-s − 64·28-s − 24·29-s + 16·31-s − 16·32-s − 32·34-s + 64·36-s + 32·39-s − 128·42-s − 24·43-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 + 8/3·9-s − 9.23·12-s − 2.21·13-s + 8.55·14-s + 15/4·16-s + 1.94·17-s − 7.54·18-s + 6.98·21-s − 1.66·23-s + 9.79·24-s + 6.27·26-s − 2.30·27-s − 12.0·28-s − 4.45·29-s + 2.87·31-s − 2.82·32-s − 5.48·34-s + 32/3·36-s + 5.12·39-s − 19.7·42-s − 3.65·43-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: \(4\)
Selberg data: \((8,\ 3^{4} \cdot 5^{8} \cdot 11^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
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$D_4\times C_2$ \( 1 + 8 T + 32 T^{2} + 136 T^{3} + 562 T^{4} + 136 p T^{5} + 32 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
17$C_2^2$ \( ( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \)
19$D_4\times C_2$ \( 1 - 52 T^{2} + 1270 T^{4} - 52 p^{2} T^{6} + p^{4} T^{8} \)
23$D_4\times C_2$ \( 1 + 8 T + 32 T^{2} + 120 T^{3} + 386 T^{4} + 120 p T^{5} + 32 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
29$D_{4}$ \( ( 1 + 12 T + 86 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )^{4} \)
37$C_2^3$ \( 1 - 2638 T^{4} + p^{4} T^{8} \)
41$D_4\times C_2$ \( 1 - 140 T^{2} + 8134 T^{4} - 140 p^{2} T^{6} + p^{4} T^{8} \)
43$D_4\times C_2$ \( 1 + 24 T + 288 T^{2} + 2664 T^{3} + 20018 T^{4} + 2664 p T^{5} + 288 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} \)
47$D_4\times C_2$ \( 1 - 8 T + 32 T^{2} - 312 T^{3} + 2978 T^{4} - 312 p T^{5} + 32 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
53$C_2^3$ \( 1 - 4174 T^{4} + p^{4} T^{8} \)
59$D_{4}$ \( ( 1 + 16 T + 150 T^{2} + 16 p T^{3} + p^{2} T^{4} )^{2} \)
61$C_2$ \( ( 1 + 6 T + p T^{2} )^{4} \)
67$D_4\times C_2$ \( 1 - 16 T + 128 T^{2} - 1520 T^{3} + 17266 T^{4} - 1520 p T^{5} + 128 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \)
71$D_4\times C_2$ \( 1 - 188 T^{2} + 16870 T^{4} - 188 p^{2} T^{6} + p^{4} T^{8} \)
73$D_4\times C_2$ \( 1 + 24 T + 288 T^{2} + 3384 T^{3} + 35138 T^{4} + 3384 p T^{5} + 288 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} \)
79$D_4\times C_2$ \( 1 - 292 T^{2} + 33670 T^{4} - 292 p^{2} T^{6} + p^{4} T^{8} \)
83$C_2^3$ \( 1 + 3122 T^{4} + p^{4} T^{8} \)
89$D_{4}$ \( ( 1 + 8 T + 66 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \)
97$D_4\times C_2$ \( 1 + 16 T + 128 T^{2} + 1040 T^{3} + 7426 T^{4} + 1040 p T^{5} + 128 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \)
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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

−7.82289622495460753491461933346, −7.75612097278575539315135710905, −7.23306289524559818203552815036, −7.16675097688764421200617147607, −6.89891476177341669924924367060, −6.76285651129293338829877432958, −6.54894670463704422511346021277, −6.14860700254316074314769674939, −6.03659367746316756295989950079, −5.94739104380483733478839876967, −5.53310771678437330488790039778, −5.50819196362044970294005649537, −5.41148103519868336719008969864, −4.59715974562372813863394394353, −4.54554643233111924371935159047, −4.50334314142555376617213983347, −3.74578817383660621165462428931, −3.56323149791529223244264881330, −3.21988890744765116859114285565, −2.97551159935104208565083682461, −2.89930760906162402174209397690, −2.09194441834005791006075524131, −2.02228626215407900731236182579, −1.41783574205398329806618711309, −1.06736866903146212394470344015, 0, 0, 0, 0, 1.06736866903146212394470344015, 1.41783574205398329806618711309, 2.02228626215407900731236182579, 2.09194441834005791006075524131, 2.89930760906162402174209397690, 2.97551159935104208565083682461, 3.21988890744765116859114285565, 3.56323149791529223244264881330, 3.74578817383660621165462428931, 4.50334314142555376617213983347, 4.54554643233111924371935159047, 4.59715974562372813863394394353, 5.41148103519868336719008969864, 5.50819196362044970294005649537, 5.53310771678437330488790039778, 5.94739104380483733478839876967, 6.03659367746316756295989950079, 6.14860700254316074314769674939, 6.54894670463704422511346021277, 6.76285651129293338829877432958, 6.89891476177341669924924367060, 7.16675097688764421200617147607, 7.23306289524559818203552815036, 7.75612097278575539315135710905, 7.82289622495460753491461933346

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