Properties

Label 8-75e4-1.1-c4e4-0-2
Degree $8$
Conductor $31640625$
Sign $1$
Analytic cond. $3612.62$
Root an. cond. $2.78437$
Motivic weight $4$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 36·4-s + 62·9-s + 460·16-s + 1.38e3·19-s − 12·31-s − 2.23e3·36-s − 1.64e3·49-s + 1.46e3·61-s + 4.32e3·64-s − 4.99e4·76-s − 1.80e4·79-s − 2.71e3·81-s + 8.54e3·109-s + 5.57e4·121-s + 432·124-s + ⋯
L(s)  = 1  − 9/4·4-s + 0.765·9-s + 1.79·16-s + 3.84·19-s − 0.0124·31-s − 1.72·36-s − 0.685·49-s + 0.394·61-s + 1.05·64-s − 8.65·76-s − 2.89·79-s − 0.414·81-s + 0.719·109-s + 3.80·121-s + 0.0280·124-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 31640625 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31640625 ^{s/2} \, \Gamma_{\C}(s+2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(31640625\)    =    \(3^{4} \cdot 5^{8}\)
Sign: $1$
Analytic conductor: \(3612.62\)
Root analytic conductor: \(2.78437\)
Motivic weight: \(4\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 31640625,\ (\ :2, 2, 2, 2),\ 1)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.170983357\)
\(L(\frac12)\) \(\approx\) \(1.170983357\)
\(L(3)\) 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$C_2^2$ \( 1 - 62 T^{2} + p^{8} T^{4} \)
5 \( 1 \)
good2$C_2^2$ \( ( 1 + 9 p T^{2} + p^{8} T^{4} )^{2} \)
7$C_2^2$ \( ( 1 + 823 T^{2} + p^{8} T^{4} )^{2} \)
11$C_2^2$ \( ( 1 - 27882 T^{2} + p^{8} T^{4} )^{2} \)
13$C_2^2$ \( ( 1 - 54097 T^{2} + p^{8} T^{4} )^{2} \)
17$C_2^2$ \( ( 1 - 84342 T^{2} + p^{8} T^{4} )^{2} \)
19$C_2$ \( ( 1 - 347 T + p^{4} T^{2} )^{4} \)
23$C_2^2$ \( ( 1 + 135818 T^{2} + p^{8} T^{4} )^{2} \)
29$C_2^2$ \( ( 1 - 673962 T^{2} + p^{8} T^{4} )^{2} \)
31$C_2$ \( ( 1 + 3 T + p^{4} T^{2} )^{4} \)
37$C_2^2$ \( ( 1 + 1224578 T^{2} + p^{8} T^{4} )^{2} \)
41$C_2^2$ \( ( 1 - 778122 T^{2} + p^{8} T^{4} )^{2} \)
43$C_2^2$ \( ( 1 - 4661977 T^{2} + p^{8} T^{4} )^{2} \)
47$C_2^2$ \( ( 1 + 6315138 T^{2} + p^{8} T^{4} )^{2} \)
53$C_2^2$ \( ( 1 + 15482538 T^{2} + p^{8} T^{4} )^{2} \)
59$C_2^2$ \( ( 1 - 16148322 T^{2} + p^{8} T^{4} )^{2} \)
61$C_2$ \( ( 1 - 367 T + p^{4} T^{2} )^{4} \)
67$C_2^2$ \( ( 1 - 35307017 T^{2} + p^{8} T^{4} )^{2} \)
71$C_2^2$ \( ( 1 - 50586762 T^{2} + p^{8} T^{4} )^{2} \)
73$C_2^2$ \( ( 1 - 8215582 T^{2} + p^{8} T^{4} )^{2} \)
79$C_2$ \( ( 1 + 4518 T + p^{4} T^{2} )^{4} \)
83$C_2^2$ \( ( 1 + 94817858 T^{2} + p^{8} T^{4} )^{2} \)
89$C_2^2$ \( ( 1 - 60166082 T^{2} + p^{8} T^{4} )^{2} \)
97$C_2^2$ \( ( 1 - 156492337 T^{2} + p^{8} T^{4} )^{2} \)
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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

−9.964977926912374806378125830323, −9.652154400290129247651475536708, −9.475999219441450549447490853224, −9.074624958589834715938401335435, −9.036391122014420159079514639684, −8.388450111245436512223305929057, −8.368886039982189503289412710463, −7.946002375030898920338204182173, −7.42913343835506134465190126042, −7.18881247949815483387407832301, −7.13726506352008042507277646918, −6.54491062523391985134797494794, −5.84420716437730339796876095712, −5.63314441918681202755397825221, −5.39200465484646165348668446127, −4.83680899117963355090906076145, −4.55076002819090054139816451457, −4.52984503101195930057900111482, −3.69637233612528062985180036292, −3.51031890928252072505269840007, −3.09585574787466342820340403112, −2.35233516062154170447978707397, −1.33833724505261647974810617410, −1.06232945521576007928313552857, −0.35099594927095634669640289743, 0.35099594927095634669640289743, 1.06232945521576007928313552857, 1.33833724505261647974810617410, 2.35233516062154170447978707397, 3.09585574787466342820340403112, 3.51031890928252072505269840007, 3.69637233612528062985180036292, 4.52984503101195930057900111482, 4.55076002819090054139816451457, 4.83680899117963355090906076145, 5.39200465484646165348668446127, 5.63314441918681202755397825221, 5.84420716437730339796876095712, 6.54491062523391985134797494794, 7.13726506352008042507277646918, 7.18881247949815483387407832301, 7.42913343835506134465190126042, 7.946002375030898920338204182173, 8.368886039982189503289412710463, 8.388450111245436512223305929057, 9.036391122014420159079514639684, 9.074624958589834715938401335435, 9.475999219441450549447490853224, 9.652154400290129247651475536708, 9.964977926912374806378125830323

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