Properties

Label 8-1078e4-1.1-c1e4-0-8
Degree $8$
Conductor $1.350\times 10^{12}$
Sign $1$
Analytic cond. $5490.14$
Root an. cond. $2.93391$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + 4-s − 2·8-s + 4·9-s + 2·11-s − 4·16-s + 8·18-s + 4·22-s + 4·23-s + 8·25-s − 24·29-s − 2·32-s + 4·36-s + 4·37-s + 16·43-s + 2·44-s + 8·46-s + 16·50-s + 24·53-s − 48·58-s + 3·64-s + 4·67-s − 40·71-s − 8·72-s + 8·74-s + 16·79-s + 9·81-s + ⋯
L(s)  = 1  + 1.41·2-s + 1/2·4-s − 0.707·8-s + 4/3·9-s + 0.603·11-s − 16-s + 1.88·18-s + 0.852·22-s + 0.834·23-s + 8/5·25-s − 4.45·29-s − 0.353·32-s + 2/3·36-s + 0.657·37-s + 2.43·43-s + 0.301·44-s + 1.17·46-s + 2.26·50-s + 3.29·53-s − 6.30·58-s + 3/8·64-s + 0.488·67-s − 4.74·71-s − 0.942·72-s + 0.929·74-s + 1.80·79-s + 81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 7^{8} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 7^{8} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 7^{8} \cdot 11^{4}\)
Sign: $1$
Analytic conductor: \(5490.14\)
Root analytic conductor: \(2.93391\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1078} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{4} \cdot 7^{8} \cdot 11^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(7.099227553\)
\(L(\frac12)\) \(\approx\) \(7.099227553\)
\(L(\frac{3}{2})\) 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)$
bad2$C_2$ \( ( 1 - T + T^{2} )^{2} \)
7 \( 1 \)
11$C_2$ \( ( 1 - T + T^{2} )^{2} \)
good3$C_2^3$ \( 1 - 4 T^{2} + 7 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} \)
5$C_2^3$ \( 1 - 8 T^{2} + 39 T^{4} - 8 p^{2} T^{6} + p^{4} T^{8} \)
13$C_2^2$ \( ( 1 + 18 T^{2} + p^{2} T^{4} )^{2} \)
17$C_2^3$ \( 1 - 2 T^{2} - 285 T^{4} - 2 p^{2} T^{6} + p^{4} T^{8} \)
19$C_2^3$ \( 1 - 30 T^{2} + 539 T^{4} - 30 p^{2} T^{6} + p^{4} T^{8} \)
23$C_2^2$ \( ( 1 - 2 T - 19 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \)
29$C_2$ \( ( 1 + 6 T + p T^{2} )^{4} \)
31$C_2^3$ \( 1 - 60 T^{2} + 2639 T^{4} - 60 p^{2} T^{6} + p^{4} T^{8} \)
37$C_2^2$ \( ( 1 - 2 T - 33 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \)
41$C_2^2$ \( ( 1 + 50 T^{2} + p^{2} T^{4} )^{2} \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )^{4} \)
47$C_2^3$ \( 1 - 76 T^{2} + 3567 T^{4} - 76 p^{2} T^{6} + p^{4} T^{8} \)
53$C_2^2$ \( ( 1 - 12 T + 91 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \)
59$C_2^3$ \( 1 - 100 T^{2} + 6519 T^{4} - 100 p^{2} T^{6} + p^{4} T^{8} \)
61$C_2^3$ \( 1 - 90 T^{2} + 4379 T^{4} - 90 p^{2} T^{6} + p^{4} T^{8} \)
67$C_2^2$ \( ( 1 - 2 T - 63 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \)
71$C_2$ \( ( 1 + 10 T + p T^{2} )^{4} \)
73$C_2^3$ \( 1 + 54 T^{2} - 2413 T^{4} + 54 p^{2} T^{6} + p^{4} T^{8} \)
79$C_2^2$ \( ( 1 - 8 T - 15 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \)
83$C_2^2$ \( ( 1 + 158 T^{2} + p^{2} T^{4} )^{2} \)
89$C_2^3$ \( 1 - 128 T^{2} + 8463 T^{4} - 128 p^{2} T^{6} + p^{4} T^{8} \)
97$C_2^2$ \( ( 1 - 48 T^{2} + p^{2} T^{4} )^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−6.98599038472824670738269954011, −6.87814900858078444822581643881, −6.60033486625472391882394591366, −6.41612098810812933989532338711, −5.99810773635683109634343046609, −5.86302115455856576529990933302, −5.59123617697526498404469089651, −5.54364405496541873922930200555, −5.26135419763845999217246132757, −5.06790717818188946222630863935, −4.70370248902019579022030748297, −4.35159464175674344870093384405, −4.22470240037869523938341830626, −4.21907368570579354291815534310, −3.78314046931653629738575030484, −3.66286346539957141566501866204, −3.47508202460065744691953011093, −2.90454780051418567248627330187, −2.82963158174833831468337548854, −2.47782074971000954998203089855, −2.08252570246507862304766255653, −1.73414405094382841272524904422, −1.43254125869397259724395104169, −0.939337805131261358351192474322, −0.50556243447685142525612914078, 0.50556243447685142525612914078, 0.939337805131261358351192474322, 1.43254125869397259724395104169, 1.73414405094382841272524904422, 2.08252570246507862304766255653, 2.47782074971000954998203089855, 2.82963158174833831468337548854, 2.90454780051418567248627330187, 3.47508202460065744691953011093, 3.66286346539957141566501866204, 3.78314046931653629738575030484, 4.21907368570579354291815534310, 4.22470240037869523938341830626, 4.35159464175674344870093384405, 4.70370248902019579022030748297, 5.06790717818188946222630863935, 5.26135419763845999217246132757, 5.54364405496541873922930200555, 5.59123617697526498404469089651, 5.86302115455856576529990933302, 5.99810773635683109634343046609, 6.41612098810812933989532338711, 6.60033486625472391882394591366, 6.87814900858078444822581643881, 6.98599038472824670738269954011

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