Properties

Label 8-21e8-1.1-c3e4-0-6
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  − 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: \(0\)
Selberg data: \((8,\ 3^{8} \cdot 7^{8} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )\)

Particular Values

\(L(2)\) \(\approx\) \(4.376057354\)
\(L(\frac12)\) \(\approx\) \(4.376057354\)
\(L(\frac{5}{2})\) 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

−7.65198067657847716597994820688, −7.40621977890869333021944868174, −7.12998304207022536890470059098, −6.76355076043563224085339990358, −6.61655896011836958560343290966, −6.39718756061717913115349679072, −6.17290848830734419058793573686, −5.63425702340553180024569511130, −5.60071182589537325431796709793, −5.17196766508970463786798420243, −4.95194526305508095153798467798, −4.85817914300657666144091404028, −4.81875797720936963741585383860, −3.86176807054527622007335400082, −3.79444637563315078394793756403, −3.73878281905455488927684908677, −3.63580062224486940137302919919, −3.02051222280658579342969778407, −2.90433292223298311564753552400, −2.04167440146916461123435943099, −2.04149999653354838319977043441, −1.36650983530787149558143635750, −1.03932504939199851000751641690, −0.55675928207389041642844818240, −0.54263788537754972647402796842, 0.54263788537754972647402796842, 0.55675928207389041642844818240, 1.03932504939199851000751641690, 1.36650983530787149558143635750, 2.04149999653354838319977043441, 2.04167440146916461123435943099, 2.90433292223298311564753552400, 3.02051222280658579342969778407, 3.63580062224486940137302919919, 3.73878281905455488927684908677, 3.79444637563315078394793756403, 3.86176807054527622007335400082, 4.81875797720936963741585383860, 4.85817914300657666144091404028, 4.95194526305508095153798467798, 5.17196766508970463786798420243, 5.60071182589537325431796709793, 5.63425702340553180024569511130, 6.17290848830734419058793573686, 6.39718756061717913115349679072, 6.61655896011836958560343290966, 6.76355076043563224085339990358, 7.12998304207022536890470059098, 7.40621977890869333021944868174, 7.65198067657847716597994820688

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