Properties

Label 8-825e4-1.1-c3e4-0-1
Degree $8$
Conductor $463250390625$
Sign $1$
Analytic cond. $5.61409\times 10^{6}$
Root an. cond. $6.97686$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s − 12·3-s − 21·4-s − 24·6-s + 11·7-s − 52·8-s + 90·9-s − 44·11-s + 252·12-s − 25·13-s + 22·14-s + 213·16-s − 85·17-s + 180·18-s + 118·19-s − 132·21-s − 88·22-s + 10·23-s + 624·24-s − 50·26-s − 540·27-s − 231·28-s + 251·29-s − 135·31-s + 630·32-s + 528·33-s − 170·34-s + ⋯
L(s)  = 1  + 0.707·2-s − 2.30·3-s − 2.62·4-s − 1.63·6-s + 0.593·7-s − 2.29·8-s + 10/3·9-s − 1.20·11-s + 6.06·12-s − 0.533·13-s + 0.419·14-s + 3.32·16-s − 1.21·17-s + 2.35·18-s + 1.42·19-s − 1.37·21-s − 0.852·22-s + 0.0906·23-s + 5.30·24-s − 0.377·26-s − 3.84·27-s − 1.55·28-s + 1.60·29-s − 0.782·31-s + 3.48·32-s + 2.78·33-s − 0.857·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(3^{4} \cdot 5^{8} \cdot 11^{4}\)
Sign: $1$
Analytic conductor: \(5.61409\times 10^{6}\)
Root analytic conductor: \(6.97686\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{825} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 3^{4} \cdot 5^{8} \cdot 11^{4} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )\)

Particular Values

\(L(2)\) \(\approx\) \(0.5565234647\)
\(L(\frac12)\) \(\approx\) \(0.5565234647\)
\(L(\frac{5}{2})\) 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$C_1$ \( ( 1 + p T )^{4} \)
5 \( 1 \)
11$C_1$ \( ( 1 + p T )^{4} \)
good2$D_{4}$ \( ( 1 - T + 3 p^{2} T^{2} - p^{3} T^{3} + p^{6} T^{4} )^{2} \)
7$C_2 \wr C_2\wr C_2$ \( 1 - 11 T + 5 p^{2} T^{2} - 4272 T^{3} + 235846 T^{4} - 4272 p^{3} T^{5} + 5 p^{8} T^{6} - 11 p^{9} T^{7} + p^{12} T^{8} \)
13$C_2 \wr C_2\wr C_2$ \( 1 + 25 T + 2848 T^{2} - 89705 T^{3} + 1153550 T^{4} - 89705 p^{3} T^{5} + 2848 p^{6} T^{6} + 25 p^{9} T^{7} + p^{12} T^{8} \)
17$C_2 \wr C_2\wr C_2$ \( 1 + 5 p T + 17959 T^{2} + 63382 p T^{3} + 126876108 T^{4} + 63382 p^{4} T^{5} + 17959 p^{6} T^{6} + 5 p^{10} T^{7} + p^{12} T^{8} \)
19$C_2 \wr C_2\wr C_2$ \( 1 - 118 T + 23772 T^{2} - 1971120 T^{3} + 225214933 T^{4} - 1971120 p^{3} T^{5} + 23772 p^{6} T^{6} - 118 p^{9} T^{7} + p^{12} T^{8} \)
23$C_2 \wr C_2\wr C_2$ \( 1 - 10 T + 19600 T^{2} + 1389080 T^{3} + 201166953 T^{4} + 1389080 p^{3} T^{5} + 19600 p^{6} T^{6} - 10 p^{9} T^{7} + p^{12} T^{8} \)
29$C_2 \wr C_2\wr C_2$ \( 1 - 251 T + 64738 T^{2} - 9591241 T^{3} + 1925819610 T^{4} - 9591241 p^{3} T^{5} + 64738 p^{6} T^{6} - 251 p^{9} T^{7} + p^{12} T^{8} \)
31$C_2 \wr C_2\wr C_2$ \( 1 + 135 T + 73642 T^{2} + 11799675 T^{3} + 2705005482 T^{4} + 11799675 p^{3} T^{5} + 73642 p^{6} T^{6} + 135 p^{9} T^{7} + p^{12} T^{8} \)
37$C_2 \wr C_2\wr C_2$ \( 1 + 419 T + 203005 T^{2} + 49657538 T^{3} + 14552591126 T^{4} + 49657538 p^{3} T^{5} + 203005 p^{6} T^{6} + 419 p^{9} T^{7} + p^{12} T^{8} \)
41$C_2 \wr C_2\wr C_2$ \( 1 + 103 T + 193405 T^{2} + 16488354 T^{3} + 18502936946 T^{4} + 16488354 p^{3} T^{5} + 193405 p^{6} T^{6} + 103 p^{9} T^{7} + p^{12} T^{8} \)
43$C_2 \wr C_2\wr C_2$ \( 1 + 15 T + 175040 T^{2} + 43959 T^{3} + 15986121134 T^{4} + 43959 p^{3} T^{5} + 175040 p^{6} T^{6} + 15 p^{9} T^{7} + p^{12} T^{8} \)
47$C_2 \wr C_2\wr C_2$ \( 1 + 665 T + 375451 T^{2} + 147542792 T^{3} + 57422466252 T^{4} + 147542792 p^{3} T^{5} + 375451 p^{6} T^{6} + 665 p^{9} T^{7} + p^{12} T^{8} \)
53$C_2 \wr C_2\wr C_2$ \( 1 - 116 T + 240584 T^{2} - 1497452 T^{3} + 30580910270 T^{4} - 1497452 p^{3} T^{5} + 240584 p^{6} T^{6} - 116 p^{9} T^{7} + p^{12} T^{8} \)
59$C_2 \wr C_2\wr C_2$ \( 1 - 951 T + 746257 T^{2} - 351355448 T^{3} + 184211670670 T^{4} - 351355448 p^{3} T^{5} + 746257 p^{6} T^{6} - 951 p^{9} T^{7} + p^{12} T^{8} \)
61$C_2 \wr C_2\wr C_2$ \( 1 + 350 T + 481192 T^{2} + 204604298 T^{3} + 125591896574 T^{4} + 204604298 p^{3} T^{5} + 481192 p^{6} T^{6} + 350 p^{9} T^{7} + p^{12} T^{8} \)
67$C_2 \wr C_2\wr C_2$ \( 1 + 266 T + 252240 T^{2} + 289246466 T^{3} + 93419783054 T^{4} + 289246466 p^{3} T^{5} + 252240 p^{6} T^{6} + 266 p^{9} T^{7} + p^{12} T^{8} \)
71$C_2 \wr C_2\wr C_2$ \( 1 - 1526 T + 2164088 T^{2} - 1770899240 T^{3} + 1299777873953 T^{4} - 1770899240 p^{3} T^{5} + 2164088 p^{6} T^{6} - 1526 p^{9} T^{7} + p^{12} T^{8} \)
73$C_2 \wr C_2\wr C_2$ \( 1 - 566 T - 231032 T^{2} + 82463206 T^{3} + 143348281550 T^{4} + 82463206 p^{3} T^{5} - 231032 p^{6} T^{6} - 566 p^{9} T^{7} + p^{12} T^{8} \)
79$C_2 \wr C_2\wr C_2$ \( 1 - 2567 T + 3969287 T^{2} - 4195920960 T^{3} + 3375898676528 T^{4} - 4195920960 p^{3} T^{5} + 3969287 p^{6} T^{6} - 2567 p^{9} T^{7} + p^{12} T^{8} \)
83$C_2 \wr C_2\wr C_2$ \( 1 - 961 T + 1800414 T^{2} - 1174919601 T^{3} + 1350021543298 T^{4} - 1174919601 p^{3} T^{5} + 1800414 p^{6} T^{6} - 961 p^{9} T^{7} + p^{12} T^{8} \)
89$C_2 \wr C_2\wr C_2$ \( 1 - 1053 T + 2481236 T^{2} - 1650076467 T^{3} + 2370212988070 T^{4} - 1650076467 p^{3} T^{5} + 2481236 p^{6} T^{6} - 1053 p^{9} T^{7} + p^{12} T^{8} \)
97$C_2 \wr C_2\wr C_2$ \( 1 - 172 T + 1942834 T^{2} - 351210592 T^{3} + 1907414271035 T^{4} - 351210592 p^{3} T^{5} + 1942834 p^{6} T^{6} - 172 p^{9} T^{7} + p^{12} T^{8} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−6.82802380318815842873543999698, −6.56420888180348765246567858192, −6.33966172896733375702714884569, −6.17186495490971305214614518981, −5.92239498824741225317861466140, −5.44761629864712419670404346748, −5.23950233873534650174201224221, −5.21844043984418814862974077306, −5.15752270681188957744070231407, −4.75221179719072633600753777248, −4.66842211640194932730591594963, −4.64063382901171336684551352692, −4.41662534728476167997437677875, −3.65786041916406184672783970382, −3.57511417708584157318285151875, −3.52338111775632256817265532671, −3.40809174806615419809050695564, −2.54533564620533475532002339786, −2.28132211055306567908884272537, −1.93292772264029760708719137861, −1.61359967550335082546408433461, −0.818194219280161136849130072491, −0.816261344864138847409443634594, −0.49986061756838714250869943604, −0.22766649383122167304307133668, 0.22766649383122167304307133668, 0.49986061756838714250869943604, 0.816261344864138847409443634594, 0.818194219280161136849130072491, 1.61359967550335082546408433461, 1.93292772264029760708719137861, 2.28132211055306567908884272537, 2.54533564620533475532002339786, 3.40809174806615419809050695564, 3.52338111775632256817265532671, 3.57511417708584157318285151875, 3.65786041916406184672783970382, 4.41662534728476167997437677875, 4.64063382901171336684551352692, 4.66842211640194932730591594963, 4.75221179719072633600753777248, 5.15752270681188957744070231407, 5.21844043984418814862974077306, 5.23950233873534650174201224221, 5.44761629864712419670404346748, 5.92239498824741225317861466140, 6.17186495490971305214614518981, 6.33966172896733375702714884569, 6.56420888180348765246567858192, 6.82802380318815842873543999698

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