↑

Properties

Label 8-75e4-1.1-c13e4-0-2
Degree $8$
Conductor $31640625$
Sign $1$
Analytic cond. $4.18336\times 10^{7}$
Root an. cond. $8.96789$
Motivic weight $13$
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  − 4.13e3·4-s − 1.06e6·9-s + 1.34e6·11-s − 6.97e7·16-s − 5.12e8·19-s + 9.45e9·29-s − 1.19e10·31-s + 4.39e9·36-s + 3.05e10·41-s − 5.55e9·44-s − 3.59e10·49-s + 1.85e12·59-s + 3.58e11·61-s + 3.69e11·64-s − 1.56e12·71-s + 2.11e12·76-s + 1.42e12·79-s + 8.47e11·81-s − 6.54e12·89-s − 1.42e12·99-s + 4.26e12·101-s − 3.43e12·109-s − 3.90e13·116-s − 8.94e13·121-s + 4.94e13·124-s + 127-s + 131-s + ⋯
L(s)  = 1  − 0.504·4-s − 2/3·9-s + 0.228·11-s − 1.03·16-s − 2.49·19-s + 2.95·29-s − 2.42·31-s + 0.336·36-s + 1.00·41-s − 0.115·44-s − 0.371·49-s + 5.72·59-s + 0.891·61-s + 0.672·64-s − 1.45·71-s + 1.26·76-s + 0.660·79-s + 1/3·81-s − 1.39·89-s − 0.152·99-s + 0.399·101-s − 0.195·109-s − 1.48·116-s − 2.59·121-s + 1.22·124-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 31640625 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31640625 ^{s/2} \, \Gamma_{\C}(s+13/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(31640625\)    =    \(3^{4} \cdot 5^{8}\)
Sign: $1$
Analytic conductor: \(4.18336\times 10^{7}\)
Root analytic conductor: \(8.96789\)
Motivic weight: \(13\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 31640625,\ (\ :13/2, 13/2, 13/2, 13/2),\ 1)\)

Particular Values

\(L(7)\) \(\approx\) \(1.187508231\)
\(L(\frac12)\) \(\approx\) \(1.187508231\)
\(L(\frac{15}{2})\) 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$C_2$ \( ( 1 + p^{12} T^{2} )^{2} \)
5 \( 1 \)
good2$D_4\times C_2$ \( 1 + 1033 p^{2} T^{2} + 84777 p^{10} T^{4} + 1033 p^{28} T^{6} + p^{52} T^{8} \)
7$D_4\times C_2$ \( 1 + 35981606116 T^{2} + \)\(38\!\cdots\!22\)\( p^{2} T^{4} + 35981606116 p^{26} T^{6} + p^{52} T^{8} \)
11$D_{4}$ \( ( 1 - 61128 p T + 375287075254 p^{2} T^{2} - 61128 p^{14} T^{3} + p^{26} T^{4} )^{2} \)
13$D_4\times C_2$ \( 1 - 771642046573292 T^{2} + \)\(28\!\cdots\!38\)\( T^{4} - 771642046573292 p^{26} T^{6} + p^{52} T^{8} \)
17$D_4\times C_2$ \( 1 - 35915196231405500 T^{2} + \)\(51\!\cdots\!38\)\( T^{4} - 35915196231405500 p^{26} T^{6} + p^{52} T^{8} \)
19$D_{4}$ \( ( 1 + 256293544 T + 90096185084470998 T^{2} + 256293544 p^{13} T^{3} + p^{26} T^{4} )^{2} \)
23$D_4\times C_2$ \( 1 - 50529716660648452 p T^{2} + \)\(66\!\cdots\!58\)\( T^{4} - 50529716660648452 p^{27} T^{6} + p^{52} T^{8} \)
29$D_{4}$ \( ( 1 - 4728475332 T + 25449630283285666078 T^{2} - 4728475332 p^{13} T^{3} + p^{26} T^{4} )^{2} \)
31$D_{4}$ \( ( 1 + 5982551648 T + 36024474609054776382 T^{2} + 5982551648 p^{13} T^{3} + p^{26} T^{4} )^{2} \)
37$D_4\times C_2$ \( 1 - \)\(58\!\cdots\!84\)\( T^{2} + \)\(20\!\cdots\!78\)\( T^{4} - \)\(58\!\cdots\!84\)\( p^{26} T^{6} + p^{52} T^{8} \)
41$D_{4}$ \( ( 1 - 15258974292 T + \)\(17\!\cdots\!82\)\( T^{2} - 15258974292 p^{13} T^{3} + p^{26} T^{4} )^{2} \)
43$D_4\times C_2$ \( 1 - \)\(33\!\cdots\!00\)\( T^{2} + \)\(85\!\cdots\!98\)\( T^{4} - \)\(33\!\cdots\!00\)\( p^{26} T^{6} + p^{52} T^{8} \)
47$D_4\times C_2$ \( 1 - \)\(25\!\cdots\!20\)\( T^{2} + \)\(20\!\cdots\!58\)\( T^{4} - \)\(25\!\cdots\!20\)\( p^{26} T^{6} + p^{52} T^{8} \)
53$D_4\times C_2$ \( 1 - \)\(17\!\cdots\!36\)\( T^{2} - \)\(11\!\cdots\!02\)\( T^{4} - \)\(17\!\cdots\!36\)\( p^{26} T^{6} + p^{52} T^{8} \)
59$D_{4}$ \( ( 1 - 927820824264 T + \)\(42\!\cdots\!38\)\( T^{2} - 927820824264 p^{13} T^{3} + p^{26} T^{4} )^{2} \)
61$D_{4}$ \( ( 1 - 179395461340 T + \)\(28\!\cdots\!98\)\( T^{2} - 179395461340 p^{13} T^{3} + p^{26} T^{4} )^{2} \)
67$D_4\times C_2$ \( 1 - \)\(63\!\cdots\!60\)\( T^{2} + \)\(38\!\cdots\!38\)\( T^{4} - \)\(63\!\cdots\!60\)\( p^{26} T^{6} + p^{52} T^{8} \)
71$D_{4}$ \( ( 1 + 784458549936 T + \)\(20\!\cdots\!46\)\( T^{2} + 784458549936 p^{13} T^{3} + p^{26} T^{4} )^{2} \)
73$D_4\times C_2$ \( 1 - \)\(31\!\cdots\!56\)\( T^{2} + \)\(50\!\cdots\!18\)\( T^{4} - \)\(31\!\cdots\!56\)\( p^{26} T^{6} + p^{52} T^{8} \)
79$D_{4}$ \( ( 1 - 714025470080 T + \)\(94\!\cdots\!78\)\( T^{2} - 714025470080 p^{13} T^{3} + p^{26} T^{4} )^{2} \)
83$D_4\times C_2$ \( 1 - \)\(24\!\cdots\!92\)\( T^{2} + \)\(31\!\cdots\!18\)\( T^{4} - \)\(24\!\cdots\!92\)\( p^{26} T^{6} + p^{52} T^{8} \)
89$D_{4}$ \( ( 1 + 3270178701684 T + \)\(37\!\cdots\!58\)\( T^{2} + 3270178701684 p^{13} T^{3} + p^{26} T^{4} )^{2} \)
97$D_4\times C_2$ \( 1 - \)\(17\!\cdots\!20\)\( T^{2} + \)\(14\!\cdots\!58\)\( T^{4} - \)\(17\!\cdots\!20\)\( p^{26} T^{6} + p^{52} T^{8} \)
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

−8.322017761234796768018636582583, −7.71416770068803470025740163683, −7.68509782100681749551359413863, −6.92896717970439402264044276704, −6.77653625956670508921585943102, −6.65518324970301608533997184786, −6.46946758020706914068766844028, −5.75650366963266915659764654747, −5.73593329293709570564809877531, −5.27392946277965139773220613960, −5.04628835235307705458478743556, −4.51304804596275107159738265371, −4.40974821753607535949157944723, −4.01031251693983256362961873298, −3.68459975371308723749412718452, −3.61044221533999554886704877215, −2.71083562475859173118088744383, −2.62540905565505041299773500837, −2.41458766851824877309279190074, −2.03304563761081471606049248325, −1.69423500244761172314085811146, −1.05565465652215812339707582826, −0.945794738770135653839103832137, −0.38713473256666342427274485244, −0.19582934546571657711785476552, 0.19582934546571657711785476552, 0.38713473256666342427274485244, 0.945794738770135653839103832137, 1.05565465652215812339707582826, 1.69423500244761172314085811146, 2.03304563761081471606049248325, 2.41458766851824877309279190074, 2.62540905565505041299773500837, 2.71083562475859173118088744383, 3.61044221533999554886704877215, 3.68459975371308723749412718452, 4.01031251693983256362961873298, 4.40974821753607535949157944723, 4.51304804596275107159738265371, 5.04628835235307705458478743556, 5.27392946277965139773220613960, 5.73593329293709570564809877531, 5.75650366963266915659764654747, 6.46946758020706914068766844028, 6.65518324970301608533997184786, 6.77653625956670508921585943102, 6.92896717970439402264044276704, 7.68509782100681749551359413863, 7.71416770068803470025740163683, 8.322017761234796768018636582583

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