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Properties

Label 8-1805e4-1.1-c1e4-0-5
Degree $8$
Conductor $1.061\times 10^{13}$
Sign $1$
Analytic cond. $43153.6$
Root an. cond. $3.79644$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $4$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 3-s − 2·4-s + 4·5-s + 6-s − 11·7-s + 8-s − 3·9-s − 4·10-s + 2·12-s + 2·13-s + 11·14-s − 4·15-s + 16-s − 7·17-s + 3·18-s − 8·20-s + 11·21-s − 11·23-s − 24-s + 10·25-s − 2·26-s + 9·27-s + 22·28-s + 15·29-s + 4·30-s + 31-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s − 4-s + 1.78·5-s + 0.408·6-s − 4.15·7-s + 0.353·8-s − 9-s − 1.26·10-s + 0.577·12-s + 0.554·13-s + 2.93·14-s − 1.03·15-s + 1/4·16-s − 1.69·17-s + 0.707·18-s − 1.78·20-s + 2.40·21-s − 2.29·23-s − 0.204·24-s + 2·25-s − 0.392·26-s + 1.73·27-s + 4.15·28-s + 2.78·29-s + 0.730·30-s + 0.179·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{4} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{4} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(5^{4} \cdot 19^{8}\)
Sign: $1$
Analytic conductor: \(43153.6\)
Root analytic conductor: \(3.79644\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(4\)
Selberg data: \((8,\ 5^{4} \cdot 19^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

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)$
bad5$C_1$ \( ( 1 - T )^{4} \)
19 \( 1 \)
good2$C_2 \wr C_2\wr C_2$ \( 1 + T + 3 T^{2} + p^{2} T^{3} + p^{3} T^{4} + p^{3} T^{5} + 3 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \)
3$C_2 \wr C_2\wr C_2$ \( 1 + T + 4 T^{2} - 2 T^{3} + 7 T^{4} - 2 p T^{5} + 4 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \)
7$C_2 \wr C_2\wr C_2$ \( 1 + 11 T + 68 T^{2} + 284 T^{3} + 873 T^{4} + 284 p T^{5} + 68 p^{2} T^{6} + 11 p^{3} T^{7} + p^{4} T^{8} \)
11$C_2 \wr C_2\wr C_2$ \( 1 + 35 T^{2} - 10 T^{3} + 527 T^{4} - 10 p T^{5} + 35 p^{2} T^{6} + p^{4} T^{8} \)
13$C_2 \wr C_2\wr C_2$ \( 1 - 2 T + 20 T^{2} - 100 T^{3} + 253 T^{4} - 100 p T^{5} + 20 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
17$C_2 \wr C_2\wr C_2$ \( 1 + 7 T + 43 T^{2} + 101 T^{3} + 448 T^{4} + 101 p T^{5} + 43 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} \)
23$C_2 \wr C_2\wr C_2$ \( 1 + 11 T + 94 T^{2} + 478 T^{3} + 2557 T^{4} + 478 p T^{5} + 94 p^{2} T^{6} + 11 p^{3} T^{7} + p^{4} T^{8} \)
29$C_2 \wr C_2\wr C_2$ \( 1 - 15 T + 165 T^{2} - 1255 T^{3} + 268 p T^{4} - 1255 p T^{5} + 165 p^{2} T^{6} - 15 p^{3} T^{7} + p^{4} T^{8} \)
31$C_2 \wr C_2\wr C_2$ \( 1 - T + 64 T^{2} - 96 T^{3} + 2705 T^{4} - 96 p T^{5} + 64 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \)
37$C_2 \wr C_2\wr C_2$ \( 1 - 11 T + 178 T^{2} - 1204 T^{3} + 10333 T^{4} - 1204 p T^{5} + 178 p^{2} T^{6} - 11 p^{3} T^{7} + p^{4} T^{8} \)
41$C_2 \wr C_2\wr C_2$ \( 1 + 22 T + 264 T^{2} + 2100 T^{3} + 14225 T^{4} + 2100 p T^{5} + 264 p^{2} T^{6} + 22 p^{3} T^{7} + p^{4} T^{8} \)
43$C_2 \wr C_2\wr C_2$ \( 1 + 26 T + 397 T^{2} + 94 p T^{3} + 30823 T^{4} + 94 p^{2} T^{5} + 397 p^{2} T^{6} + 26 p^{3} T^{7} + p^{4} T^{8} \)
47$C_2 \wr C_2\wr C_2$ \( 1 + 26 T + 388 T^{2} + 3904 T^{3} + 30373 T^{4} + 3904 p T^{5} + 388 p^{2} T^{6} + 26 p^{3} T^{7} + p^{4} T^{8} \)
53$C_2 \wr C_2\wr C_2$ \( 1 - 16 T + 237 T^{2} - 2012 T^{3} + 17613 T^{4} - 2012 p T^{5} + 237 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \)
59$C_2 \wr C_2\wr C_2$ \( 1 - 10 T + 195 T^{2} - 1420 T^{3} + 16877 T^{4} - 1420 p T^{5} + 195 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \)
61$C_2 \wr C_2\wr C_2$ \( 1 - 2 T + 113 T^{2} + 566 T^{3} + 4600 T^{4} + 566 p T^{5} + 113 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
67$C_2 \wr C_2\wr C_2$ \( 1 + 3 T + 158 T^{2} + 524 T^{3} + 14583 T^{4} + 524 p T^{5} + 158 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \)
71$C_2 \wr C_2\wr C_2$ \( 1 + 18 T + 384 T^{2} + 4000 T^{3} + 44465 T^{4} + 4000 p T^{5} + 384 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \)
73$C_2 \wr C_2\wr C_2$ \( 1 + 24 T + 442 T^{2} + 5168 T^{3} + 51803 T^{4} + 5168 p T^{5} + 442 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} \)
79$C_2 \wr C_2\wr C_2$ \( 1 + 30 T + 637 T^{2} + 8570 T^{3} + 90548 T^{4} + 8570 p T^{5} + 637 p^{2} T^{6} + 30 p^{3} T^{7} + p^{4} T^{8} \)
83$C_2 \wr C_2\wr C_2$ \( 1 + 12 T + 327 T^{2} + 2692 T^{3} + 40508 T^{4} + 2692 p T^{5} + 327 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \)
89$C_2 \wr C_2\wr C_2$ \( 1 + 9 T + 4 p T^{2} + 2390 T^{3} + 47525 T^{4} + 2390 p T^{5} + 4 p^{3} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \)
97$C_2 \wr C_2\wr C_2$ \( 1 + 19 T + 340 T^{2} + 4300 T^{3} + 46303 T^{4} + 4300 p T^{5} + 340 p^{2} T^{6} + 19 p^{3} T^{7} + p^{4} T^{8} \)
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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

−6.87949053178501435282410532768, −6.73437144441975477482652207910, −6.59223318398197395373562962525, −6.26082470991179082995167469003, −6.24256115155707740896494620499, −6.12031500498692171299919701468, −5.90800583887361252115700949291, −5.67789132021935583309302244838, −5.55493649976042875383061307124, −4.96298457855662896642007588427, −4.86966431752158648384806420625, −4.83143647020547364163938877286, −4.55188130547868824581503151623, −4.10708829601966542975994261094, −3.92468130988421657728969581405, −3.67953200170138527747880619613, −3.12967804780164720681069809144, −3.11010337654967640435408234786, −3.10581063877843075820460278495, −2.72768537561347978172728469928, −2.64652863544610167503576718447, −2.14910618556348512450075785163, −1.76492921063241349373842219227, −1.38814385731444096100905722486, −1.08720259256496384442333576610, 0, 0, 0, 0, 1.08720259256496384442333576610, 1.38814385731444096100905722486, 1.76492921063241349373842219227, 2.14910618556348512450075785163, 2.64652863544610167503576718447, 2.72768537561347978172728469928, 3.10581063877843075820460278495, 3.11010337654967640435408234786, 3.12967804780164720681069809144, 3.67953200170138527747880619613, 3.92468130988421657728969581405, 4.10708829601966542975994261094, 4.55188130547868824581503151623, 4.83143647020547364163938877286, 4.86966431752158648384806420625, 4.96298457855662896642007588427, 5.55493649976042875383061307124, 5.67789132021935583309302244838, 5.90800583887361252115700949291, 6.12031500498692171299919701468, 6.24256115155707740896494620499, 6.26082470991179082995167469003, 6.59223318398197395373562962525, 6.73437144441975477482652207910, 6.87949053178501435282410532768

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