↑

Properties

Label 8-10e8-1.1-c11e4-0-0
Degree $8$
Conductor $100000000$
Sign $1$
Analytic cond. $3.48514\times 10^{7}$
Root an. cond. $8.76551$
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  + 3.10e5·9-s + 5.96e5·11-s − 3.88e7·19-s − 8.36e7·29-s + 1.24e8·31-s − 1.21e9·41-s + 3.29e9·49-s − 7.51e9·59-s + 9.84e9·61-s − 8.67e10·71-s + 1.09e11·79-s + 1.86e10·81-s − 7.37e9·89-s + 1.85e11·99-s + 1.90e10·101-s − 3.51e11·109-s − 8.61e11·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2.71e12·169-s + ⋯
L(s)  = 1  + 1.75·9-s + 1.11·11-s − 3.60·19-s − 0.757·29-s + 0.780·31-s − 1.64·41-s + 1.66·49-s − 1.36·59-s + 1.49·61-s − 5.70·71-s + 4.00·79-s + 0.593·81-s − 0.140·89-s + 1.95·99-s + 0.180·101-s − 2.18·109-s − 3.02·121-s + 1.51·169-s − 6.31·171-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(12-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+11/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{8} \cdot 5^{8}\)
Sign: $1$
Analytic conductor: \(3.48514\times 10^{7}\)
Root analytic conductor: \(8.76551\)
Motivic weight: \(11\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{8} \cdot 5^{8} ,\ ( \ : 11/2, 11/2, 11/2, 11/2 ),\ 1 )\)

Particular Values

\(L(6)\) \(\approx\) \(2.709853305\)
\(L(\frac12)\) \(\approx\) \(2.709853305\)
\(L(\frac{13}{2})\) 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 \( 1 \)
5 \( 1 \)
good3$D_4\times C_2$ \( 1 - 310556 T^{2} + 960822742 p^{4} T^{4} - 310556 p^{22} T^{6} + p^{44} T^{8} \)
7$D_4\times C_2$ \( 1 - 3297745100 T^{2} + 214545541333728102 p^{2} T^{4} - 3297745100 p^{22} T^{6} + p^{44} T^{8} \)
11$D_{4}$ \( ( 1 - 27120 p T + 564442123222 T^{2} - 27120 p^{12} T^{3} + p^{22} T^{4} )^{2} \)
13$D_4\times C_2$ \( 1 - 2713317032396 T^{2} + \)\(55\!\cdots\!42\)\( T^{4} - 2713317032396 p^{22} T^{6} + p^{44} T^{8} \)
17$D_4\times C_2$ \( 1 - 5095774828732 p T^{2} + \)\(38\!\cdots\!62\)\( T^{4} - 5095774828732 p^{23} T^{6} + p^{44} T^{8} \)
19$D_{4}$ \( ( 1 + 19439368 T + 284381272377894 T^{2} + 19439368 p^{11} T^{3} + p^{22} T^{4} )^{2} \)
23$D_4\times C_2$ \( 1 - 2231382077679500 T^{2} + \)\(57\!\cdots\!02\)\( p^{2} T^{4} - 2231382077679500 p^{22} T^{6} + p^{44} T^{8} \)
29$D_{4}$ \( ( 1 + 41841708 T + 3630990695891374 T^{2} + 41841708 p^{11} T^{3} + p^{22} T^{4} )^{2} \)
31$D_{4}$ \( ( 1 - 62230792 T - 13999135099203522 T^{2} - 62230792 p^{11} T^{3} + p^{22} T^{4} )^{2} \)
37$D_4\times C_2$ \( 1 - 245850534239205740 T^{2} + \)\(25\!\cdots\!38\)\( T^{4} - 245850534239205740 p^{22} T^{6} + p^{44} T^{8} \)
41$D_{4}$ \( ( 1 + 608419068 T + 1119141666750688438 T^{2} + 608419068 p^{11} T^{3} + p^{22} T^{4} )^{2} \)
43$D_4\times C_2$ \( 1 - 1936176957438272828 T^{2} + \)\(19\!\cdots\!94\)\( T^{4} - 1936176957438272828 p^{22} T^{6} + p^{44} T^{8} \)
47$D_4\times C_2$ \( 1 - 4735615217502786284 T^{2} + \)\(12\!\cdots\!82\)\( T^{4} - 4735615217502786284 p^{22} T^{6} + p^{44} T^{8} \)
53$D_4\times C_2$ \( 1 - 14956042439370797420 T^{2} + \)\(13\!\cdots\!18\)\( T^{4} - 14956042439370797420 p^{22} T^{6} + p^{44} T^{8} \)
59$D_{4}$ \( ( 1 + 3756433896 T + 50973876172267502422 T^{2} + 3756433896 p^{11} T^{3} + p^{22} T^{4} )^{2} \)
61$D_{4}$ \( ( 1 - 4923703564 T + 12712509351121558446 T^{2} - 4923703564 p^{11} T^{3} + p^{22} T^{4} )^{2} \)
67$D_4\times C_2$ \( 1 - \)\(45\!\cdots\!20\)\( T^{2} + \)\(80\!\cdots\!78\)\( T^{4} - \)\(45\!\cdots\!20\)\( p^{22} T^{6} + p^{44} T^{8} \)
71$D_{4}$ \( ( 1 + 43352162664 T + \)\(91\!\cdots\!66\)\( T^{2} + 43352162664 p^{11} T^{3} + p^{22} T^{4} )^{2} \)
73$D_4\times C_2$ \( 1 - \)\(64\!\cdots\!56\)\( T^{2} + \)\(27\!\cdots\!42\)\( T^{4} - \)\(64\!\cdots\!56\)\( p^{22} T^{6} + p^{44} T^{8} \)
79$D_{4}$ \( ( 1 - 54799425296 T + \)\(19\!\cdots\!62\)\( T^{2} - 54799425296 p^{11} T^{3} + p^{22} T^{4} )^{2} \)
83$D_4\times C_2$ \( 1 - \)\(50\!\cdots\!40\)\( T^{2} + \)\(96\!\cdots\!78\)\( T^{4} - \)\(50\!\cdots\!40\)\( p^{22} T^{6} + p^{44} T^{8} \)
89$D_{4}$ \( ( 1 + 3688968372 T + \)\(27\!\cdots\!74\)\( T^{2} + 3688968372 p^{11} T^{3} + p^{22} T^{4} )^{2} \)
97$D_4\times C_2$ \( 1 - \)\(21\!\cdots\!60\)\( T^{2} + \)\(21\!\cdots\!18\)\( T^{4} - \)\(21\!\cdots\!60\)\( p^{22} T^{6} + p^{44} T^{8} \)
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

−8.055774604831662441333290145262, −7.74962840686120511824504749241, −7.43327392551241213508710714167, −6.94235299780274028245855753163, −6.70337496050786545945899482755, −6.63575563147756528132709651928, −6.38549862605137555717875046466, −6.09193695430236954294566089399, −5.53398899184562027096963625139, −5.38176781178730068070355928825, −4.84675419991143342583499925900, −4.49623312177754966031432285609, −4.19521547028338060262649976766, −4.12713457469784255122708863787, −3.92908781316879397110745689672, −3.49108262496584601394297489072, −2.95042370363198756735192383498, −2.59481095417784780201883048493, −2.26550628270040570842631705702, −1.83033360643703335032291080451, −1.58144860620679341273152262887, −1.43798883828376200575127089437, −1.01711793716218467518392554030, −0.42894135060175122116852715259, −0.23202621285325640285977349418, 0.23202621285325640285977349418, 0.42894135060175122116852715259, 1.01711793716218467518392554030, 1.43798883828376200575127089437, 1.58144860620679341273152262887, 1.83033360643703335032291080451, 2.26550628270040570842631705702, 2.59481095417784780201883048493, 2.95042370363198756735192383498, 3.49108262496584601394297489072, 3.92908781316879397110745689672, 4.12713457469784255122708863787, 4.19521547028338060262649976766, 4.49623312177754966031432285609, 4.84675419991143342583499925900, 5.38176781178730068070355928825, 5.53398899184562027096963625139, 6.09193695430236954294566089399, 6.38549862605137555717875046466, 6.63575563147756528132709651928, 6.70337496050786545945899482755, 6.94235299780274028245855753163, 7.43327392551241213508710714167, 7.74962840686120511824504749241, 8.055774604831662441333290145262

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