↑

Properties

Label 8-50e4-1.1-c11e4-0-0
Degree $8$
Conductor $6250000$
Sign $1$
Analytic cond. $2.17821\times 10^{6}$
Root an. cond. $6.19815$
Motivic weight $11$
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  − 2.04e3·4-s + 2.00e5·9-s − 1.99e5·11-s + 3.14e6·16-s − 2.34e7·19-s − 2.46e8·29-s + 2.45e8·31-s − 4.10e8·36-s + 3.09e9·41-s + 4.07e8·44-s + 5.51e9·49-s − 2.14e10·59-s + 3.45e10·61-s − 4.29e9·64-s − 1.36e10·71-s + 4.81e10·76-s − 4.97e10·79-s − 3.17e10·81-s + 2.86e10·89-s − 3.99e10·99-s − 4.33e11·101-s − 2.41e11·109-s + 5.03e11·116-s − 7.41e11·121-s − 5.03e11·124-s + 127-s + 131-s + ⋯
L(s)  = 1  − 4-s + 1.13·9-s − 0.372·11-s + 3/4·16-s − 2.17·19-s − 2.22·29-s + 1.54·31-s − 1.13·36-s + 4.16·41-s + 0.372·44-s + 2.79·49-s − 3.90·59-s + 5.23·61-s − 1/2·64-s − 0.897·71-s + 2.17·76-s − 1.81·79-s − 1.01·81-s + 0.543·89-s − 0.422·99-s − 4.10·101-s − 1.50·109-s + 2.22·116-s − 2.59·121-s − 1.54·124-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6250000 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6250000 ^{s/2} \, \Gamma_{\C}(s+11/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(6250000\)    =    \(2^{4} \cdot 5^{8}\)
Sign: $1$
Analytic conductor: \(2.17821\times 10^{6}\)
Root analytic conductor: \(6.19815\)
Motivic weight: \(11\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 6250000,\ (\ :11/2, 11/2, 11/2, 11/2),\ 1)\)

Particular Values

\(L(6)\) \(\approx\) \(0.7843073155\)
\(L(\frac12)\) \(\approx\) \(0.7843073155\)
\(L(\frac{13}{2})\) 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)$
bad2$C_2$ \( ( 1 + p^{10} T^{2} )^{2} \)
5 \( 1 \)
good3$D_4\times C_2$ \( 1 - 200570 T^{2} + 889198603 p^{4} T^{4} - 200570 p^{22} T^{6} + p^{44} T^{8} \)
7$D_4\times C_2$ \( 1 - 112637300 p^{2} T^{2} + 5871716297361798 p^{4} T^{4} - 112637300 p^{24} T^{6} + p^{44} T^{8} \)
11$D_{4}$ \( ( 1 + 99576 T + 385381175941 T^{2} + 99576 p^{11} T^{3} + p^{22} T^{4} )^{2} \)
13$D_4\times C_2$ \( 1 - 4823100556340 T^{2} + \)\(11\!\cdots\!38\)\( T^{4} - 4823100556340 p^{22} T^{6} + p^{44} T^{8} \)
17$D_4\times C_2$ \( 1 + 14739248380270 T^{2} + \)\(10\!\cdots\!03\)\( T^{4} + 14739248380270 p^{22} T^{6} + p^{44} T^{8} \)
19$D_{4}$ \( ( 1 + 11745160 T + 107539119308613 T^{2} + 11745160 p^{11} T^{3} + p^{22} T^{4} )^{2} \)
23$D_4\times C_2$ \( 1 + 86656460712940 T^{2} - \)\(33\!\cdots\!42\)\( T^{4} + 86656460712940 p^{22} T^{6} + p^{44} T^{8} \)
29$D_{4}$ \( ( 1 + 123023640 T + 27732658283523658 T^{2} + 123023640 p^{11} T^{3} + p^{22} T^{4} )^{2} \)
31$D_{4}$ \( ( 1 - 122902864 T + 12992750360035386 T^{2} - 122902864 p^{11} T^{3} + p^{22} T^{4} )^{2} \)
37$D_4\times C_2$ \( 1 - 175511944622664140 T^{2} + \)\(64\!\cdots\!38\)\( T^{4} - 175511944622664140 p^{22} T^{6} + p^{44} T^{8} \)
41$D_{4}$ \( ( 1 - 1546691034 T + 41110690182650531 p T^{2} - 1546691034 p^{11} T^{3} + p^{22} T^{4} )^{2} \)
43$D_4\times C_2$ \( 1 - 2302454154542269100 T^{2} + \)\(30\!\cdots\!98\)\( T^{4} - 2302454154542269100 p^{22} T^{6} + p^{44} T^{8} \)
47$D_4\times C_2$ \( 1 - 98342046278555420 T^{2} - \)\(12\!\cdots\!82\)\( T^{4} - 98342046278555420 p^{22} T^{6} + p^{44} T^{8} \)
53$D_4\times C_2$ \( 1 - 29195825090170733420 T^{2} + \)\(37\!\cdots\!18\)\( T^{4} - 29195825090170733420 p^{22} T^{6} + p^{44} T^{8} \)
59$D_{4}$ \( ( 1 + 10715159280 T + 87440751827151373318 T^{2} + 10715159280 p^{11} T^{3} + p^{22} T^{4} )^{2} \)
61$D_{4}$ \( ( 1 - 17250789724 T + \)\(15\!\cdots\!66\)\( T^{2} - 17250789724 p^{11} T^{3} + p^{22} T^{4} )^{2} \)
67$D_4\times C_2$ \( 1 - \)\(20\!\cdots\!30\)\( T^{2} + \)\(36\!\cdots\!03\)\( T^{4} - \)\(20\!\cdots\!30\)\( p^{22} T^{6} + p^{44} T^{8} \)
71$D_{4}$ \( ( 1 + 6825023256 T + \)\(47\!\cdots\!26\)\( T^{2} + 6825023256 p^{11} T^{3} + p^{22} T^{4} )^{2} \)
73$D_4\times C_2$ \( 1 - \)\(50\!\cdots\!10\)\( T^{2} + \)\(13\!\cdots\!83\)\( T^{4} - \)\(50\!\cdots\!10\)\( p^{22} T^{6} + p^{44} T^{8} \)
79$D_{4}$ \( ( 1 + 24885895240 T + \)\(15\!\cdots\!58\)\( T^{2} + 24885895240 p^{11} T^{3} + p^{22} T^{4} )^{2} \)
83$D_4\times C_2$ \( 1 - \)\(45\!\cdots\!30\)\( T^{2} + \)\(85\!\cdots\!03\)\( T^{4} - \)\(45\!\cdots\!30\)\( p^{22} T^{6} + p^{44} T^{8} \)
89$D_{4}$ \( ( 1 - 14317932930 T + \)\(24\!\cdots\!03\)\( T^{2} - 14317932930 p^{11} T^{3} + p^{22} T^{4} )^{2} \)
97$D_4\times C_2$ \( 1 - \)\(26\!\cdots\!20\)\( T^{2} + \)\(27\!\cdots\!18\)\( T^{4} - \)\(26\!\cdots\!20\)\( p^{22} T^{6} + p^{44} T^{8} \)
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

−9.274706391361799559500835826242, −8.964939668753329553799891836080, −8.369947311780720605250852004018, −8.280515907195027282817925116500, −7.87395332527299899410599189105, −7.45315206580001437994140627580, −7.35619570820090589964003124215, −6.78033103545357336965464666413, −6.57174919668040104153173918604, −5.97345735452259642306900874706, −5.82387392044964153429234378725, −5.34516658127219180723278494097, −5.18641750264760581097123802806, −4.29109155289447476004849116124, −4.20180487566652658495495684370, −4.13373624038569336227038174166, −3.99373798759946033534367622054, −3.08742128549138986341501616189, −2.52038317653359542711533599039, −2.50298513422215794101637293285, −1.95252657043239782282373116662, −1.21952263035078481888215219096, −1.21574795263540872019598168314, −0.59191842832132747201657637092, −0.15134815171727371722387075296, 0.15134815171727371722387075296, 0.59191842832132747201657637092, 1.21574795263540872019598168314, 1.21952263035078481888215219096, 1.95252657043239782282373116662, 2.50298513422215794101637293285, 2.52038317653359542711533599039, 3.08742128549138986341501616189, 3.99373798759946033534367622054, 4.13373624038569336227038174166, 4.20180487566652658495495684370, 4.29109155289447476004849116124, 5.18641750264760581097123802806, 5.34516658127219180723278494097, 5.82387392044964153429234378725, 5.97345735452259642306900874706, 6.57174919668040104153173918604, 6.78033103545357336965464666413, 7.35619570820090589964003124215, 7.45315206580001437994140627580, 7.87395332527299899410599189105, 8.280515907195027282817925116500, 8.369947311780720605250852004018, 8.964939668753329553799891836080, 9.274706391361799559500835826242

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