Properties

Label 8-21e8-1.1-c3e4-0-9
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 $4$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 13·4-s − 102·13-s + 47·16-s − 222·19-s − 67·25-s − 220·31-s − 374·37-s − 838·43-s + 1.32e3·52-s − 1.33e3·61-s + 143·64-s + 1.89e3·67-s − 1.75e3·73-s + 2.88e3·76-s + 8·79-s − 3.01e3·97-s + 871·100-s − 2.04e3·103-s − 1.01e3·109-s − 1.38e3·121-s + 2.86e3·124-s + 127-s + 131-s + 137-s + 139-s + 4.86e3·148-s + 149-s + ⋯
L(s)  = 1  − 1.62·4-s − 2.17·13-s + 0.734·16-s − 2.68·19-s − 0.535·25-s − 1.27·31-s − 1.66·37-s − 2.97·43-s + 3.53·52-s − 2.79·61-s + 0.279·64-s + 3.44·67-s − 2.80·73-s + 4.35·76-s + 0.0113·79-s − 3.15·97-s + 0.870·100-s − 1.95·103-s − 0.887·109-s − 1.04·121-s + 2.07·124-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 2.70·148-s + 0.000549·149-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: \(4\)
Selberg data: \((8,\ 3^{8} \cdot 7^{8} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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$C_2^2 \wr C_2$ \( 1 + 13 T^{2} + 61 p T^{4} + 13 p^{6} T^{6} + p^{12} T^{8} \)
5$C_2^2 \wr C_2$ \( 1 + 67 T^{2} + 18428 T^{4} + 67 p^{6} T^{6} + p^{12} T^{8} \)
11$C_2^2 \wr C_2$ \( 1 + 1387 T^{2} + 837200 T^{4} + 1387 p^{6} T^{6} + p^{12} T^{8} \)
13$D_{4}$ \( ( 1 + 51 T + 4996 T^{2} + 51 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
17$C_2^2 \wr C_2$ \( 1 + 4256 T^{2} + 51629310 T^{4} + 4256 p^{6} T^{6} + p^{12} T^{8} \)
19$D_{4}$ \( ( 1 + 111 T + 10960 T^{2} + 111 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
23$C_2^2 \wr C_2$ \( 1 + 8216 T^{2} - 51412626 T^{4} + 8216 p^{6} T^{6} + p^{12} T^{8} \)
29$C_2^2 \wr C_2$ \( 1 + 59371 T^{2} + 1802370044 T^{4} + 59371 p^{6} T^{6} + p^{12} T^{8} \)
31$D_{4}$ \( ( 1 + 110 T + 1397 p T^{2} + 110 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
37$D_{4}$ \( ( 1 + 187 T + 79892 T^{2} + 187 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
41$C_2^2 \wr C_2$ \( 1 + 85540 T^{2} + 8435951270 T^{4} + 85540 p^{6} T^{6} + p^{12} T^{8} \)
43$D_{4}$ \( ( 1 + 419 T + 167730 T^{2} + 419 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
47$C_2^2 \wr C_2$ \( 1 + 279692 T^{2} + 41075202726 T^{4} + 279692 p^{6} T^{6} + p^{12} T^{8} \)
53$C_2^2 \wr C_2$ \( 1 + 574307 T^{2} + 126700633692 T^{4} + 574307 p^{6} T^{6} + p^{12} T^{8} \)
59$C_2^2 \wr C_2$ \( 1 + 615851 T^{2} + 176286397968 T^{4} + 615851 p^{6} T^{6} + p^{12} T^{8} \)
61$D_{4}$ \( ( 1 + 666 T + 462754 T^{2} + 666 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
67$D_{4}$ \( ( 1 - 945 T + 807364 T^{2} - 945 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
71$C_2^2 \wr C_2$ \( 1 + 874364 T^{2} + 392544633894 T^{4} + 874364 p^{6} T^{6} + p^{12} T^{8} \)
73$D_{4}$ \( ( 1 + 875 T + 967076 T^{2} + 875 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
79$D_{4}$ \( ( 1 - 4 T + 900969 T^{2} - 4 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
83$C_2^2 \wr C_2$ \( 1 + 1482323 T^{2} + 1051851285456 T^{4} + 1482323 p^{6} T^{6} + p^{12} T^{8} \)
89$C_2^2 \wr C_2$ \( 1 + 627904 T^{2} + 1046910984158 T^{4} + 627904 p^{6} T^{6} + p^{12} T^{8} \)
97$D_{4}$ \( ( 1 + 1505 T + 1992044 T^{2} + 1505 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
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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

−8.356418030785471403749077742703, −7.912308298284962636032479727096, −7.62831977428277707497391886396, −7.37375341585540121241272294002, −7.05502429206257969010356626111, −6.91444832273275243292503653981, −6.65468622735242876350767124597, −6.44773557013951570054889898880, −6.11489608697863692898591598325, −5.77681276532736521518640849569, −5.33014805024167671628966362962, −5.25078624997037342326830925106, −5.12425670975547098620064535641, −4.72734250238702833409624480412, −4.39255978509377782188902152584, −4.34021411218410451899138453955, −4.12511813104430031466636897007, −3.53252452496670269995780672539, −3.52696004030330274073052593649, −3.03659359450312523268893677263, −2.54814497758050854209640791558, −2.25400953870055639832462913043, −2.03441783252030420224266247137, −1.51822188586692182582240420571, −1.23583670498177209247174643099, 0, 0, 0, 0, 1.23583670498177209247174643099, 1.51822188586692182582240420571, 2.03441783252030420224266247137, 2.25400953870055639832462913043, 2.54814497758050854209640791558, 3.03659359450312523268893677263, 3.52696004030330274073052593649, 3.53252452496670269995780672539, 4.12511813104430031466636897007, 4.34021411218410451899138453955, 4.39255978509377782188902152584, 4.72734250238702833409624480412, 5.12425670975547098620064535641, 5.25078624997037342326830925106, 5.33014805024167671628966362962, 5.77681276532736521518640849569, 6.11489608697863692898591598325, 6.44773557013951570054889898880, 6.65468622735242876350767124597, 6.91444832273275243292503653981, 7.05502429206257969010356626111, 7.37375341585540121241272294002, 7.62831977428277707497391886396, 7.912308298284962636032479727096, 8.356418030785471403749077742703

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