Properties

Label 8-637e4-1.1-c3e4-0-0
Degree $8$
Conductor $164648481361$
Sign $1$
Analytic cond. $1.99536\times 10^{6}$
Root an. cond. $6.13059$
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  − 4·2-s + 5·3-s + 5·4-s + 36·5-s − 20·6-s + 2·8-s − 31·9-s − 144·10-s − 95·11-s + 25·12-s + 52·13-s + 180·15-s − 49·16-s + 146·17-s + 124·18-s + 48·19-s + 180·20-s + 380·22-s − 121·23-s + 10·24-s + 651·25-s − 208·26-s − 214·27-s − 440·29-s − 720·30-s + 283·31-s + 184·32-s + ⋯
L(s)  = 1  − 1.41·2-s + 0.962·3-s + 5/8·4-s + 3.21·5-s − 1.36·6-s + 0.0883·8-s − 1.14·9-s − 4.55·10-s − 2.60·11-s + 0.601·12-s + 1.10·13-s + 3.09·15-s − 0.765·16-s + 2.08·17-s + 1.62·18-s + 0.579·19-s + 2.01·20-s + 3.68·22-s − 1.09·23-s + 0.0850·24-s + 5.20·25-s − 1.56·26-s − 1.52·27-s − 2.81·29-s − 4.38·30-s + 1.63·31-s + 1.01·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(7^{8} \cdot 13^{4}\)
Sign: $1$
Analytic conductor: \(1.99536\times 10^{6}\)
Root analytic conductor: \(6.13059\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{637} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 7^{8} \cdot 13^{4} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )\)

Particular Values

\(L(2)\) \(\approx\) \(7.352106744\)
\(L(\frac12)\) \(\approx\) \(7.352106744\)
\(L(\frac{5}{2})\) 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)$
bad7 \( 1 \)
13$C_1$ \( ( 1 - p T )^{4} \)
good2$C_2 \wr S_4$ \( 1 + p^{2} T + 11 T^{2} + 11 p T^{3} + 37 p T^{4} + 11 p^{4} T^{5} + 11 p^{6} T^{6} + p^{11} T^{7} + p^{12} T^{8} \)
3$C_2 \wr S_4$ \( 1 - 5 T + 56 T^{2} - 221 T^{3} + 2014 T^{4} - 221 p^{3} T^{5} + 56 p^{6} T^{6} - 5 p^{9} T^{7} + p^{12} T^{8} \)
5$C_2 \wr S_4$ \( 1 - 36 T + 129 p T^{2} - 8694 T^{3} + 102712 T^{4} - 8694 p^{3} T^{5} + 129 p^{7} T^{6} - 36 p^{9} T^{7} + p^{12} T^{8} \)
11$C_2 \wr S_4$ \( 1 + 95 T + 7452 T^{2} + 379875 T^{3} + 16142470 T^{4} + 379875 p^{3} T^{5} + 7452 p^{6} T^{6} + 95 p^{9} T^{7} + p^{12} T^{8} \)
17$C_2 \wr S_4$ \( 1 - 146 T + 16532 T^{2} - 1364486 T^{3} + 113102822 T^{4} - 1364486 p^{3} T^{5} + 16532 p^{6} T^{6} - 146 p^{9} T^{7} + p^{12} T^{8} \)
19$C_2 \wr S_4$ \( 1 - 48 T + 22109 T^{2} - 1039926 T^{3} + 209516732 T^{4} - 1039926 p^{3} T^{5} + 22109 p^{6} T^{6} - 48 p^{9} T^{7} + p^{12} T^{8} \)
23$C_2 \wr S_4$ \( 1 + 121 T + 43427 T^{2} + 4124692 T^{3} + 758296736 T^{4} + 4124692 p^{3} T^{5} + 43427 p^{6} T^{6} + 121 p^{9} T^{7} + p^{12} T^{8} \)
29$C_2 \wr S_4$ \( 1 + 440 T + 121385 T^{2} + 24466046 T^{3} + 4246931120 T^{4} + 24466046 p^{3} T^{5} + 121385 p^{6} T^{6} + 440 p^{9} T^{7} + p^{12} T^{8} \)
31$C_2 \wr S_4$ \( 1 - 283 T + 115883 T^{2} - 22509948 T^{3} + 5128506688 T^{4} - 22509948 p^{3} T^{5} + 115883 p^{6} T^{6} - 283 p^{9} T^{7} + p^{12} T^{8} \)
37$C_2 \wr S_4$ \( 1 + 209 T + 110126 T^{2} + 34693527 T^{3} + 6353129866 T^{4} + 34693527 p^{3} T^{5} + 110126 p^{6} T^{6} + 209 p^{9} T^{7} + p^{12} T^{8} \)
41$C_2 \wr S_4$ \( 1 - 93 T + 29038 T^{2} - 7679451 T^{3} + 6599220738 T^{4} - 7679451 p^{3} T^{5} + 29038 p^{6} T^{6} - 93 p^{9} T^{7} + p^{12} T^{8} \)
43$C_2 \wr S_4$ \( 1 - 526 T + 389933 T^{2} - 128599950 T^{3} + 49380663820 T^{4} - 128599950 p^{3} T^{5} + 389933 p^{6} T^{6} - 526 p^{9} T^{7} + p^{12} T^{8} \)
47$C_2 \wr S_4$ \( 1 - 783 T + 602263 T^{2} - 253090440 T^{3} + 102444830856 T^{4} - 253090440 p^{3} T^{5} + 602263 p^{6} T^{6} - 783 p^{9} T^{7} + p^{12} T^{8} \)
53$C_2 \wr S_4$ \( 1 + 340 T + 376337 T^{2} + 45654214 T^{3} + 56508894008 T^{4} + 45654214 p^{3} T^{5} + 376337 p^{6} T^{6} + 340 p^{9} T^{7} + p^{12} T^{8} \)
59$C_2 \wr S_4$ \( 1 - 922 T + 707452 T^{2} - 393421354 T^{3} + 216277613526 T^{4} - 393421354 p^{3} T^{5} + 707452 p^{6} T^{6} - 922 p^{9} T^{7} + p^{12} T^{8} \)
61$C_2 \wr S_4$ \( 1 - 141 T + 898886 T^{2} - 95008047 T^{3} + 305023047986 T^{4} - 95008047 p^{3} T^{5} + 898886 p^{6} T^{6} - 141 p^{9} T^{7} + p^{12} T^{8} \)
67$C_2 \wr S_4$ \( 1 + 523 T + 1250888 T^{2} + 464057559 T^{3} + 570573180206 T^{4} + 464057559 p^{3} T^{5} + 1250888 p^{6} T^{6} + 523 p^{9} T^{7} + p^{12} T^{8} \)
71$C_2 \wr S_4$ \( 1 - 1468 T + 2102788 T^{2} - 1679767276 T^{3} + 1251908502678 T^{4} - 1679767276 p^{3} T^{5} + 2102788 p^{6} T^{6} - 1468 p^{9} T^{7} + p^{12} T^{8} \)
73$C_2 \wr S_4$ \( 1 - 47 T + 562529 T^{2} - 207207510 T^{3} + 173123133922 T^{4} - 207207510 p^{3} T^{5} + 562529 p^{6} T^{6} - 47 p^{9} T^{7} + p^{12} T^{8} \)
79$C_2 \wr S_4$ \( 1 - 1025 T + 2071119 T^{2} - 1439609696 T^{3} + 1542926691608 T^{4} - 1439609696 p^{3} T^{5} + 2071119 p^{6} T^{6} - 1025 p^{9} T^{7} + p^{12} T^{8} \)
83$C_2 \wr S_4$ \( 1 - 1190 T + 2309473 T^{2} - 1949369070 T^{3} + 1975771812452 T^{4} - 1949369070 p^{3} T^{5} + 2309473 p^{6} T^{6} - 1190 p^{9} T^{7} + p^{12} T^{8} \)
89$C_2 \wr S_4$ \( 1 - 2962 T + 5920469 T^{2} - 7608132226 T^{3} + 7558598583356 T^{4} - 7608132226 p^{3} T^{5} + 5920469 p^{6} T^{6} - 2962 p^{9} T^{7} + p^{12} T^{8} \)
97$C_2 \wr S_4$ \( 1 + 2715 T + 4771625 T^{2} + 5953314954 T^{3} + 6129100067190 T^{4} + 5953314954 p^{3} T^{5} + 4771625 p^{6} T^{6} + 2715 p^{9} T^{7} + p^{12} T^{8} \)
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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

−7.31663139948896536721116654219, −7.30362619629177823981523283312, −6.66817526982635892158480229725, −6.29532715256418426997262208792, −6.23913536963086960102406260523, −5.83016685641442517885654495533, −5.79739860777240498851366873930, −5.59772951830086948914602965163, −5.45365748047739460146722661382, −5.31131335856563119055053092595, −4.93205434287457061336837738118, −4.56418885231914121835207948444, −4.27571261878339588144802216775, −3.53527649087751659065401979489, −3.41008949213299738769864750537, −3.35637337814673355949900831068, −2.92408850674410124758235794130, −2.37466154066137516251455862969, −2.35532681044712597965097552126, −2.09615915652645558356068895376, −2.03543412323423528469399223909, −1.54954691305521978022400681216, −0.839961680932392632255188834272, −0.64578977229240372491271704913, −0.54517485489772946694852008339, 0.54517485489772946694852008339, 0.64578977229240372491271704913, 0.839961680932392632255188834272, 1.54954691305521978022400681216, 2.03543412323423528469399223909, 2.09615915652645558356068895376, 2.35532681044712597965097552126, 2.37466154066137516251455862969, 2.92408850674410124758235794130, 3.35637337814673355949900831068, 3.41008949213299738769864750537, 3.53527649087751659065401979489, 4.27571261878339588144802216775, 4.56418885231914121835207948444, 4.93205434287457061336837738118, 5.31131335856563119055053092595, 5.45365748047739460146722661382, 5.59772951830086948914602965163, 5.79739860777240498851366873930, 5.83016685641442517885654495533, 6.23913536963086960102406260523, 6.29532715256418426997262208792, 6.66817526982635892158480229725, 7.30362619629177823981523283312, 7.31663139948896536721116654219

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