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Properties

Label 8-77e4-1.1-c3e4-0-0
Degree $8$
Conductor $35153041$
Sign $1$
Analytic cond. $426.017$
Root an. cond. $2.13146$
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  − 2·2-s + 14·3-s − 4-s + 10·5-s − 28·6-s − 28·7-s + 4·8-s + 82·9-s − 20·10-s − 44·11-s − 14·12-s + 58·13-s + 56·14-s + 140·15-s − 51·16-s + 4·17-s − 164·18-s + 258·19-s − 10·20-s − 392·21-s + 88·22-s + 8·23-s + 56·24-s − 160·25-s − 116·26-s + 250·27-s + 28·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 2.69·3-s − 1/8·4-s + 0.894·5-s − 1.90·6-s − 1.51·7-s + 0.176·8-s + 3.03·9-s − 0.632·10-s − 1.20·11-s − 0.336·12-s + 1.23·13-s + 1.06·14-s + 2.40·15-s − 0.796·16-s + 0.0570·17-s − 2.14·18-s + 3.11·19-s − 0.111·20-s − 4.07·21-s + 0.852·22-s + 0.0725·23-s + 0.476·24-s − 1.27·25-s − 0.874·26-s + 1.78·27-s + 0.188·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 35153041 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35153041 ^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(35153041\)    =    \(7^{4} \cdot 11^{4}\)
Sign: $1$
Analytic conductor: \(426.017\)
Root analytic conductor: \(2.13146\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 35153041,\ (\ :3/2, 3/2, 3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(3.581936592\)
\(L(\frac12)\) \(\approx\) \(3.581936592\)
\(L(\frac{5}{2})\) 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$C_1$ \( ( 1 + p T )^{4} \)
11$C_1$ \( ( 1 + p T )^{4} \)
good2$C_2 \wr S_4$ \( 1 + p T + 5 T^{2} + p^{3} T^{3} + p^{6} T^{4} + p^{6} T^{5} + 5 p^{6} T^{6} + p^{10} T^{7} + p^{12} T^{8} \)
3$C_2 \wr S_4$ \( 1 - 14 T + 38 p T^{2} - 698 T^{3} + 3682 T^{4} - 698 p^{3} T^{5} + 38 p^{7} T^{6} - 14 p^{9} T^{7} + p^{12} T^{8} \)
5$C_2 \wr S_4$ \( 1 - 2 p T + 52 p T^{2} - 1102 T^{3} + 29734 T^{4} - 1102 p^{3} T^{5} + 52 p^{7} T^{6} - 2 p^{10} T^{7} + p^{12} T^{8} \)
13$C_2 \wr S_4$ \( 1 - 58 T + 3862 T^{2} - 39850 T^{3} + 2368354 T^{4} - 39850 p^{3} T^{5} + 3862 p^{6} T^{6} - 58 p^{9} T^{7} + p^{12} T^{8} \)
17$C_2 \wr S_4$ \( 1 - 4 T + 13466 T^{2} + 198500 T^{3} + 81336754 T^{4} + 198500 p^{3} T^{5} + 13466 p^{6} T^{6} - 4 p^{9} T^{7} + p^{12} T^{8} \)
19$C_2 \wr S_4$ \( 1 - 258 T + 2296 p T^{2} - 5147154 T^{3} + 489918366 T^{4} - 5147154 p^{3} T^{5} + 2296 p^{7} T^{6} - 258 p^{9} T^{7} + p^{12} T^{8} \)
23$C_2 \wr S_4$ \( 1 - 8 T + 26276 T^{2} + 1258456 T^{3} + 325878550 T^{4} + 1258456 p^{3} T^{5} + 26276 p^{6} T^{6} - 8 p^{9} T^{7} + p^{12} T^{8} \)
29$C_2 \wr S_4$ \( 1 + 396 T + 138428 T^{2} + 30598020 T^{3} + 5584930806 T^{4} + 30598020 p^{3} T^{5} + 138428 p^{6} T^{6} + 396 p^{9} T^{7} + p^{12} T^{8} \)
31$C_2 \wr S_4$ \( 1 + 56 T + 87274 T^{2} + 3513992 T^{3} + 3413701858 T^{4} + 3513992 p^{3} T^{5} + 87274 p^{6} T^{6} + 56 p^{9} T^{7} + p^{12} T^{8} \)
37$C_2 \wr S_4$ \( 1 - 84 T + 139096 T^{2} - 12117036 T^{3} + 8970963870 T^{4} - 12117036 p^{3} T^{5} + 139096 p^{6} T^{6} - 84 p^{9} T^{7} + p^{12} T^{8} \)
41$C_2 \wr S_4$ \( 1 - 52 T + 191978 T^{2} + 4000964 T^{3} + 16303189330 T^{4} + 4000964 p^{3} T^{5} + 191978 p^{6} T^{6} - 52 p^{9} T^{7} + p^{12} T^{8} \)
43$C_2 \wr S_4$ \( 1 - 408 T + 227224 T^{2} - 65489736 T^{3} + 24699468414 T^{4} - 65489736 p^{3} T^{5} + 227224 p^{6} T^{6} - 408 p^{9} T^{7} + p^{12} T^{8} \)
47$C_2 \wr S_4$ \( 1 - 8 T + 293642 T^{2} - 14787896 T^{3} + 39096224482 T^{4} - 14787896 p^{3} T^{5} + 293642 p^{6} T^{6} - 8 p^{9} T^{7} + p^{12} T^{8} \)
53$C_2 \wr S_4$ \( 1 - 624 T + 731456 T^{2} - 291124560 T^{3} + 173869150638 T^{4} - 291124560 p^{3} T^{5} + 731456 p^{6} T^{6} - 624 p^{9} T^{7} + p^{12} T^{8} \)
59$C_2 \wr S_4$ \( 1 + 238 T + 320090 T^{2} - 3740918 T^{3} + 64718281666 T^{4} - 3740918 p^{3} T^{5} + 320090 p^{6} T^{6} + 238 p^{9} T^{7} + p^{12} T^{8} \)
61$C_2 \wr S_4$ \( 1 + 162 T + 320206 T^{2} - 56573262 T^{3} + 48989538114 T^{4} - 56573262 p^{3} T^{5} + 320206 p^{6} T^{6} + 162 p^{9} T^{7} + p^{12} T^{8} \)
67$C_2 \wr S_4$ \( 1 - 20 p T + 1030948 T^{2} - 550745180 T^{3} + 298359795814 T^{4} - 550745180 p^{3} T^{5} + 1030948 p^{6} T^{6} - 20 p^{10} T^{7} + p^{12} T^{8} \)
71$C_2 \wr S_4$ \( 1 - 1788 T + 2193428 T^{2} - 1809946572 T^{3} + 1240921367286 T^{4} - 1809946572 p^{3} T^{5} + 2193428 p^{6} T^{6} - 1788 p^{9} T^{7} + p^{12} T^{8} \)
73$C_2 \wr S_4$ \( 1 - 1456 T + 1167226 T^{2} - 565357096 T^{3} + 283420832722 T^{4} - 565357096 p^{3} T^{5} + 1167226 p^{6} T^{6} - 1456 p^{9} T^{7} + p^{12} T^{8} \)
79$C_2 \wr S_4$ \( 1 + 1324 T + 2043976 T^{2} + 1611726940 T^{3} + 1469807561518 T^{4} + 1611726940 p^{3} T^{5} + 2043976 p^{6} T^{6} + 1324 p^{9} T^{7} + p^{12} T^{8} \)
83$C_2 \wr S_4$ \( 1 - 450 T + 1903832 T^{2} - 631295250 T^{3} + 1545243120894 T^{4} - 631295250 p^{3} T^{5} + 1903832 p^{6} T^{6} - 450 p^{9} T^{7} + p^{12} T^{8} \)
89$C_2 \wr S_4$ \( 1 + 3072 T + 5688284 T^{2} + 7201250304 T^{3} + 6916861622502 T^{4} + 7201250304 p^{3} T^{5} + 5688284 p^{6} T^{6} + 3072 p^{9} T^{7} + p^{12} T^{8} \)
97$C_2 \wr S_4$ \( 1 + 652 T + 1754332 T^{2} + 1157057716 T^{3} + 2404953539782 T^{4} + 1157057716 p^{3} T^{5} + 1754332 p^{6} T^{6} + 652 p^{9} T^{7} + p^{12} T^{8} \)
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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

−9.938573404440751019378374404244, −9.528906049874955755825838851889, −9.526276618024973511492510156539, −9.503148295890569814062316108485, −9.136415183947955875825588239580, −8.796999221867646801378178713084, −8.429485578433457882742053432961, −8.142710663245254033937630792022, −8.019544139203197811619475935889, −7.40839495440179040815122085971, −7.33012226363956329434309047287, −7.15151849433477564682408333868, −6.40904214762277365570525333670, −6.07986757278846757743002364832, −5.63081948969745927183828584007, −5.38063466885693824937421282006, −5.11019764685693697382838367818, −3.90473427660955481741568971391, −3.78833903313197625175139554581, −3.61307897541876531810231453317, −2.91872760787979495632881567677, −2.61758488666166272765233973829, −2.35984090381648759747012213328, −1.58216166679708247101339440373, −0.60686660060071262838453011611, 0.60686660060071262838453011611, 1.58216166679708247101339440373, 2.35984090381648759747012213328, 2.61758488666166272765233973829, 2.91872760787979495632881567677, 3.61307897541876531810231453317, 3.78833903313197625175139554581, 3.90473427660955481741568971391, 5.11019764685693697382838367818, 5.38063466885693824937421282006, 5.63081948969745927183828584007, 6.07986757278846757743002364832, 6.40904214762277365570525333670, 7.15151849433477564682408333868, 7.33012226363956329434309047287, 7.40839495440179040815122085971, 8.019544139203197811619475935889, 8.142710663245254033937630792022, 8.429485578433457882742053432961, 8.796999221867646801378178713084, 9.136415183947955875825588239580, 9.503148295890569814062316108485, 9.526276618024973511492510156539, 9.528906049874955755825838851889, 9.938573404440751019378374404244

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