↑

Properties

Label 8-3675e4-1.1-c1e4-0-16
Degree $8$
Conductor $1.824\times 10^{14}$
Sign $1$
Analytic cond. $741545.$
Root an. cond. $5.41710$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $4$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 4·3-s − 8·6-s + 2·8-s + 10·9-s − 2·13-s − 2·16-s − 2·17-s − 20·18-s − 12·19-s − 10·23-s + 8·24-s + 4·26-s + 20·27-s − 6·29-s − 8·31-s + 4·32-s + 4·34-s − 24·37-s + 24·38-s − 8·39-s + 4·41-s − 8·43-s + 20·46-s + 10·47-s − 8·48-s − 8·51-s + ⋯
L(s)  = 1  − 1.41·2-s + 2.30·3-s − 3.26·6-s + 0.707·8-s + 10/3·9-s − 0.554·13-s − 1/2·16-s − 0.485·17-s − 4.71·18-s − 2.75·19-s − 2.08·23-s + 1.63·24-s + 0.784·26-s + 3.84·27-s − 1.11·29-s − 1.43·31-s + 0.707·32-s + 0.685·34-s − 3.94·37-s + 3.89·38-s − 1.28·39-s + 0.624·41-s − 1.21·43-s + 2.94·46-s + 1.45·47-s − 1.15·48-s − 1.12·51-s + ⋯

Functional equation

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

Invariants

Degree: \(8\)
Conductor: \(3^{4} \cdot 5^{8} \cdot 7^{8}\)
Sign: $1$
Analytic conductor: \(741545.\)
Root analytic conductor: \(5.41710\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(4\)
Selberg data: \((8,\ 3^{4} \cdot 5^{8} \cdot 7^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good2$C_2 \wr S_4$ \( 1 + p T + p^{2} T^{2} + 3 p T^{3} + 5 p T^{4} + 3 p^{2} T^{5} + p^{4} T^{6} + p^{4} T^{7} + p^{4} T^{8} \)
11$C_2 \wr S_4$ \( 1 + 26 T^{2} - 14 T^{3} + 360 T^{4} - 14 p T^{5} + 26 p^{2} T^{6} + p^{4} T^{8} \)
13$C_2 \wr S_4$ \( 1 + 2 T + 24 T^{2} + 42 T^{3} + 413 T^{4} + 42 p T^{5} + 24 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
17$C_2 \wr S_4$ \( 1 + 2 T + 40 T^{2} + 80 T^{3} + 916 T^{4} + 80 p T^{5} + 40 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
19$C_2 \wr S_4$ \( 1 + 12 T + 6 p T^{2} + 724 T^{3} + 3619 T^{4} + 724 p T^{5} + 6 p^{3} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \)
23$C_2 \wr S_4$ \( 1 + 10 T + 60 T^{2} + 104 T^{3} + 196 T^{4} + 104 p T^{5} + 60 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \)
29$C_2 \wr S_4$ \( 1 + 6 T + 78 T^{2} + 332 T^{3} + 2820 T^{4} + 332 p T^{5} + 78 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
31$C_2 \wr S_4$ \( 1 + 8 T + 118 T^{2} + 648 T^{3} + 5455 T^{4} + 648 p T^{5} + 118 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
37$C_2 \wr S_4$ \( 1 + 24 T + 352 T^{2} + 3386 T^{3} + 24217 T^{4} + 3386 p T^{5} + 352 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} \)
41$C_2 \wr S_4$ \( 1 - 4 T + 114 T^{2} - 346 T^{3} + 5976 T^{4} - 346 p T^{5} + 114 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
43$C_2 \wr S_4$ \( 1 + 8 T + 140 T^{2} + 878 T^{3} + 8293 T^{4} + 878 p T^{5} + 140 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
47$C_2 \wr S_4$ \( 1 - 10 T + 204 T^{2} - 1346 T^{3} + 14638 T^{4} - 1346 p T^{5} + 204 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \)
53$C_2 \wr S_4$ \( 1 + 20 T + 332 T^{2} + 3388 T^{3} + 29670 T^{4} + 3388 p T^{5} + 332 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} \)
59$C_2 \wr S_4$ \( 1 - 2 T + 230 T^{2} - 348 T^{3} + 20188 T^{4} - 348 p T^{5} + 230 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
61$C_2 \wr S_4$ \( 1 + 8 T + 144 T^{2} + 764 T^{3} + 10626 T^{4} + 764 p T^{5} + 144 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
67$C_2 \wr S_4$ \( 1 + 6 T + 196 T^{2} + 662 T^{3} + 16381 T^{4} + 662 p T^{5} + 196 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
71$C_2 \wr S_4$ \( 1 + 14 T + 194 T^{2} + 1648 T^{3} + 14264 T^{4} + 1648 p T^{5} + 194 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \)
73$C_2 \wr S_4$ \( 1 + 12 T + 228 T^{2} + 1834 T^{3} + 21241 T^{4} + 1834 p T^{5} + 228 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \)
79$C_2 \wr S_4$ \( 1 + 8 T + 162 T^{2} + 1212 T^{3} + 20195 T^{4} + 1212 p T^{5} + 162 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
83$C_2 \wr S_4$ \( 1 - 6 T + 276 T^{2} - 1256 T^{3} + 32400 T^{4} - 1256 p T^{5} + 276 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
89$C_2 \wr S_4$ \( 1 - 8 T + 162 T^{2} - 1150 T^{3} + 140 p T^{4} - 1150 p T^{5} + 162 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
97$C_2 \wr S_4$ \( 1 - 2 T + 380 T^{2} - 566 T^{3} + 54898 T^{4} - 566 p T^{5} + 380 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
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

−6.39469707225489388743156889802, −6.37471582000510707836496825371, −6.21241636935230905127244747441, −5.87593052574891832278288073727, −5.76838716268651223823080828739, −5.28001073138395919042217562606, −5.08448624212120699747763168331, −5.03383098039058290785453727663, −4.90555590115137837170602763790, −4.35289405227750532019247485144, −4.28394530354246107915254612402, −4.15449772192719059228750931968, −4.02105416149781718471520502500, −3.61823541569508247029507250307, −3.58129766243539706588085540788, −3.50623839435450937314198996412, −2.97873459424739741040624514773, −2.70534990639154177324241042954, −2.67143902649286571039711038335, −2.24371777492466777778070082910, −2.03819045241024911082761473782, −2.01665939436374606443979050561, −1.48873476158269063007092415890, −1.46504004314762340670932972746, −1.32382847698060557297331603862, 0, 0, 0, 0, 1.32382847698060557297331603862, 1.46504004314762340670932972746, 1.48873476158269063007092415890, 2.01665939436374606443979050561, 2.03819045241024911082761473782, 2.24371777492466777778070082910, 2.67143902649286571039711038335, 2.70534990639154177324241042954, 2.97873459424739741040624514773, 3.50623839435450937314198996412, 3.58129766243539706588085540788, 3.61823541569508247029507250307, 4.02105416149781718471520502500, 4.15449772192719059228750931968, 4.28394530354246107915254612402, 4.35289405227750532019247485144, 4.90555590115137837170602763790, 5.03383098039058290785453727663, 5.08448624212120699747763168331, 5.28001073138395919042217562606, 5.76838716268651223823080828739, 5.87593052574891832278288073727, 6.21241636935230905127244747441, 6.37471582000510707836496825371, 6.39469707225489388743156889802

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