Properties

Label 8-21e8-1.1-c3e4-0-5
Degree $8$
Conductor $37822859361$
Sign $1$
Analytic cond. $458372.$
Root an. cond. $5.10096$
Motivic weight $3$
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  + 9·4-s + 64·16-s + 250·25-s + 900·37-s + 720·43-s + 999·64-s + 1.48e3·67-s + 2.76e3·79-s + 2.25e3·100-s − 108·109-s + 1.96e3·121-s + 127-s + 131-s + 137-s + 139-s + 8.10e3·148-s + 149-s + 151-s + 157-s + 163-s + 167-s − 8.78e3·169-s + 6.48e3·172-s + 173-s + 179-s + 181-s + 191-s + ⋯
L(s)  = 1  + 9/8·4-s + 16-s + 2·25-s + 3.99·37-s + 2.55·43-s + 1.95·64-s + 2.69·67-s + 3.94·79-s + 9/4·100-s − 0.0949·109-s + 1.47·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 4.49·148-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s − 4·169-s + 2.87·172-s + 0.000439·173-s + 0.000417·179-s + 0.000410·181-s + 0.000378·191-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(3^{8} \cdot 7^{8}\)
Sign: $1$
Analytic conductor: \(458372.\)
Root analytic conductor: \(5.10096\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{441} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 3^{8} \cdot 7^{8} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )\)

Particular Values

\(L(2)\) \(\approx\) \(9.524817493\)
\(L(\frac12)\) \(\approx\) \(9.524817493\)
\(L(\frac{5}{2})\) 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 \( 1 \)
7 \( 1 \)
good2$C_2^3$ \( 1 - 9 T^{2} + 17 T^{4} - 9 p^{6} T^{6} + p^{12} T^{8} \)
5$C_2^2$ \( ( 1 - p^{3} T^{2} + p^{6} T^{4} )^{2} \)
11$C_2^3$ \( 1 - 1962 T^{2} + 2077883 T^{4} - 1962 p^{6} T^{6} + p^{12} T^{8} \)
13$C_2$ \( ( 1 + p^{3} T^{2} )^{4} \)
17$C_2^2$ \( ( 1 - p^{3} T^{2} + p^{6} T^{4} )^{2} \)
19$C_2^2$ \( ( 1 - p^{3} T^{2} + p^{6} T^{4} )^{2} \)
23$C_2^3$ \( 1 + 22734 T^{2} + 368798867 T^{4} + 22734 p^{6} T^{6} + p^{12} T^{8} \)
29$C_2^2$ \( ( 1 - 21222 T^{2} + p^{6} T^{4} )^{2} \)
31$C_2^2$ \( ( 1 - p^{3} T^{2} + p^{6} T^{4} )^{2} \)
37$C_2^2$ \( ( 1 - 450 T + 151847 T^{2} - 450 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
41$C_2$ \( ( 1 + p^{3} T^{2} )^{4} \)
43$C_2$ \( ( 1 - 180 T + p^{3} T^{2} )^{4} \)
47$C_2^2$ \( ( 1 - p^{3} T^{2} + p^{6} T^{4} )^{2} \)
53$C_2^3$ \( 1 - 50346 T^{2} - 19629641413 T^{4} - 50346 p^{6} T^{6} + p^{12} T^{8} \)
59$C_2^2$ \( ( 1 - p^{3} T^{2} + p^{6} T^{4} )^{2} \)
61$C_2^2$ \( ( 1 - p^{3} T^{2} + p^{6} T^{4} )^{2} \)
67$C_2^2$ \( ( 1 - 740 T + 246837 T^{2} - 740 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
71$C_2^2$ \( ( 1 - 242478 T^{2} + p^{6} T^{4} )^{2} \)
73$C_2^2$ \( ( 1 - p^{3} T^{2} + p^{6} T^{4} )^{2} \)
79$C_2^2$ \( ( 1 - 1384 T + 1422417 T^{2} - 1384 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
83$C_2$ \( ( 1 + p^{3} T^{2} )^{4} \)
89$C_2^2$ \( ( 1 - p^{3} T^{2} + p^{6} T^{4} )^{2} \)
97$C_2$ \( ( 1 + p^{3} T^{2} )^{4} \)
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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

−7.78869916312217894240066233442, −7.12082316789439417638092366455, −7.09592392411988801053508046220, −7.09219819162166368940436969345, −6.53973108056831443657152726811, −6.32169442387289091295253546696, −6.15816486978293473419801527833, −5.97456635154081055275876411196, −5.77983622001850314864533994394, −5.20773169922782504110776139700, −5.00202026899963173516861191815, −4.92321189606286189974128164240, −4.55410936206650880031490132760, −4.17024369324560223942935646878, −3.69044131594193917848776751902, −3.67758177154065858239470615053, −3.44311651880787863689911215572, −2.56711111153160769481502607587, −2.51967202906074956288802097086, −2.50215889982708529360717775313, −2.31005036587521560682388428959, −1.33382297849119120412243813919, −1.12173978266417370166712426020, −0.910214200376569666844401432401, −0.45574886040584609680642970686, 0.45574886040584609680642970686, 0.910214200376569666844401432401, 1.12173978266417370166712426020, 1.33382297849119120412243813919, 2.31005036587521560682388428959, 2.50215889982708529360717775313, 2.51967202906074956288802097086, 2.56711111153160769481502607587, 3.44311651880787863689911215572, 3.67758177154065858239470615053, 3.69044131594193917848776751902, 4.17024369324560223942935646878, 4.55410936206650880031490132760, 4.92321189606286189974128164240, 5.00202026899963173516861191815, 5.20773169922782504110776139700, 5.77983622001850314864533994394, 5.97456635154081055275876411196, 6.15816486978293473419801527833, 6.32169442387289091295253546696, 6.53973108056831443657152726811, 7.09219819162166368940436969345, 7.09592392411988801053508046220, 7.12082316789439417638092366455, 7.78869916312217894240066233442

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