↑

Properties

Label 8-6825e4-1.1-c1e4-0-1
Degree $8$
Conductor $2.170\times 10^{15}$
Sign $1$
Analytic cond. $8.82102\times 10^{6}$
Root an. cond. $7.38226$
Motivic weight $1$
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 − 4·3-s − 3·4-s − 8·6-s − 4·7-s − 10·8-s + 10·9-s − 4·11-s + 12·12-s − 4·13-s − 8·14-s + 2·17-s + 20·18-s − 2·19-s + 16·21-s − 8·22-s + 8·23-s + 40·24-s − 8·26-s − 20·27-s + 12·28-s − 6·29-s − 2·31-s + 18·32-s + 16·33-s + 4·34-s − 30·36-s + ⋯
L(s)  = 1  + 1.41·2-s − 2.30·3-s − 3/2·4-s − 3.26·6-s − 1.51·7-s − 3.53·8-s + 10/3·9-s − 1.20·11-s + 3.46·12-s − 1.10·13-s − 2.13·14-s + 0.485·17-s + 4.71·18-s − 0.458·19-s + 3.49·21-s − 1.70·22-s + 1.66·23-s + 8.16·24-s − 1.56·26-s − 3.84·27-s + 2.26·28-s − 1.11·29-s − 0.359·31-s + 3.18·32-s + 2.78·33-s + 0.685·34-s − 5·36-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 7^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 7^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(3^{4} \cdot 5^{8} \cdot 7^{4} \cdot 13^{4}\)
Sign: $1$
Analytic conductor: \(8.82102\times 10^{6}\)
Root analytic conductor: \(7.38226\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 3^{4} \cdot 5^{8} \cdot 7^{4} \cdot 13^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(1.388250888\)
\(L(\frac12)\) \(\approx\) \(1.388250888\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 + T )^{4} \)
5 \( 1 \)
7$C_1$ \( ( 1 + T )^{4} \)
13$C_1$ \( ( 1 + T )^{4} \)
good2$D_{4}$ \( ( 1 - T + 3 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \)
11$((C_8 : C_2):C_2):C_2$ \( 1 + 4 T + 30 T^{2} + 96 T^{3} + 479 T^{4} + 96 p T^{5} + 30 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
17$((C_8 : C_2):C_2):C_2$ \( 1 - 2 T + 42 T^{2} - 100 T^{3} + 951 T^{4} - 100 p T^{5} + 42 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
19$((C_8 : C_2):C_2):C_2$ \( 1 + 2 T + 65 T^{2} + 82 T^{3} + 91 p T^{4} + 82 p T^{5} + 65 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
23$((C_8 : C_2):C_2):C_2$ \( 1 - 8 T + 101 T^{2} - 534 T^{3} + 3569 T^{4} - 534 p T^{5} + 101 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
29$((C_8 : C_2):C_2):C_2$ \( 1 + 6 T + 117 T^{2} + 518 T^{3} + 5105 T^{4} + 518 p T^{5} + 117 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
31$((C_8 : C_2):C_2):C_2$ \( 1 + 2 T + 58 T^{2} + 264 T^{3} + 1655 T^{4} + 264 p T^{5} + 58 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
37$((C_8 : C_2):C_2):C_2$ \( 1 - 10 T + 178 T^{2} - 1140 T^{3} + 10439 T^{4} - 1140 p T^{5} + 178 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \)
41$C_2^2:C_4$ \( 1 + 154 T^{2} + 9271 T^{4} + 154 p^{2} T^{6} + p^{4} T^{8} \)
43$((C_8 : C_2):C_2):C_2$ \( 1 - 18 T + 191 T^{2} - 1664 T^{3} + 12489 T^{4} - 1664 p T^{5} + 191 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \)
47$((C_8 : C_2):C_2):C_2$ \( 1 - 12 T + 222 T^{2} - 1640 T^{3} + 16211 T^{4} - 1640 p T^{5} + 222 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \)
53$((C_8 : C_2):C_2):C_2$ \( 1 + 10 T + 202 T^{2} + 1580 T^{3} + 15799 T^{4} + 1580 p T^{5} + 202 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \)
59$((C_8 : C_2):C_2):C_2$ \( 1 - 4 T + 187 T^{2} - 692 T^{3} + 15345 T^{4} - 692 p T^{5} + 187 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
61$((C_8 : C_2):C_2):C_2$ \( 1 + 20 T + 374 T^{2} + 4000 T^{3} + 631 p T^{4} + 4000 p T^{5} + 374 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} \)
67$((C_8 : C_2):C_2):C_2$ \( 1 - 10 T + 158 T^{2} - 1640 T^{3} + 12499 T^{4} - 1640 p T^{5} + 158 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \)
71$((C_8 : C_2):C_2):C_2$ \( 1 + 14 T + 255 T^{2} + 2216 T^{3} + 24209 T^{4} + 2216 p T^{5} + 255 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \)
73$((C_8 : C_2):C_2):C_2$ \( 1 - 14 T + 318 T^{2} - 2980 T^{3} + 35571 T^{4} - 2980 p T^{5} + 318 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \)
79$((C_8 : C_2):C_2):C_2$ \( 1 + 26 T + 527 T^{2} + 6628 T^{3} + 70285 T^{4} + 6628 p T^{5} + 527 p^{2} T^{6} + 26 p^{3} T^{7} + p^{4} T^{8} \)
83$((C_8 : C_2):C_2):C_2$ \( 1 - 12 T + 171 T^{2} - 1056 T^{3} + 10029 T^{4} - 1056 p T^{5} + 171 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \)
89$((C_8 : C_2):C_2):C_2$ \( 1 - 20 T + 431 T^{2} - 5380 T^{3} + 60781 T^{4} - 5380 p T^{5} + 431 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} \)
97$((C_8 : C_2):C_2):C_2$ \( 1 - 16 T + 449 T^{2} - 4682 T^{3} + 68209 T^{4} - 4682 p T^{5} + 449 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \)
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

−5.59124267563460893155864036910, −5.25084992510186582674665771618, −5.10909401904738429748573112214, −5.10407936355553837028246038606, −5.03706162329560723232659510088, −4.54277073799381581772794460793, −4.52243365810382718000388886067, −4.44541185011148299386942370041, −4.36814024240719053193099255260, −3.79815998776423379135720769160, −3.77720038212870484144114248507, −3.71985346218120755869071960649, −3.64593083050634253322113643807, −3.03118431940881103007065324075, −2.82889362995771543324737867077, −2.76975629140544290324126523798, −2.74584740089400308056921626654, −2.20804211714064584970428923858, −1.98854654944513581473999907990, −1.53679352668078478049427331318, −1.40725985915697740650579310252, −0.65232794399704564605918304597, −0.61663502512433466467920577471, −0.47403569632589382266361033011, −0.41114356658827423294850887943, 0.41114356658827423294850887943, 0.47403569632589382266361033011, 0.61663502512433466467920577471, 0.65232794399704564605918304597, 1.40725985915697740650579310252, 1.53679352668078478049427331318, 1.98854654944513581473999907990, 2.20804211714064584970428923858, 2.74584740089400308056921626654, 2.76975629140544290324126523798, 2.82889362995771543324737867077, 3.03118431940881103007065324075, 3.64593083050634253322113643807, 3.71985346218120755869071960649, 3.77720038212870484144114248507, 3.79815998776423379135720769160, 4.36814024240719053193099255260, 4.44541185011148299386942370041, 4.52243365810382718000388886067, 4.54277073799381581772794460793, 5.03706162329560723232659510088, 5.10407936355553837028246038606, 5.10909401904738429748573112214, 5.25084992510186582674665771618, 5.59124267563460893155864036910

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