Properties

Label 8-91e8-1.1-c1e4-0-7
Degree $8$
Conductor $4.703\times 10^{15}$
Sign $1$
Analytic cond. $1.91178\times 10^{7}$
Root an. cond. $8.13167$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $4$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·2-s + 2·4-s − 20·8-s − 8·9-s + 4·11-s − 45·16-s − 32·18-s + 16·22-s − 12·23-s + 4·25-s − 16·29-s + 16·32-s − 16·36-s − 8·37-s + 12·43-s + 8·44-s − 48·46-s + 16·50-s − 12·53-s − 64·58-s + 204·64-s + 4·67-s + 24·71-s + 160·72-s − 32·74-s − 28·79-s + 30·81-s + ⋯
L(s)  = 1  + 2.82·2-s + 4-s − 7.07·8-s − 8/3·9-s + 1.20·11-s − 11.2·16-s − 7.54·18-s + 3.41·22-s − 2.50·23-s + 4/5·25-s − 2.97·29-s + 2.82·32-s − 8/3·36-s − 1.31·37-s + 1.82·43-s + 1.20·44-s − 7.07·46-s + 2.26·50-s − 1.64·53-s − 8.40·58-s + 51/2·64-s + 0.488·67-s + 2.84·71-s + 18.8·72-s − 3.71·74-s − 3.15·79-s + 10/3·81-s + ⋯

Functional equation

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

Invariants

Degree: \(8\)
Conductor: \(7^{8} \cdot 13^{8}\)
Sign: $1$
Analytic conductor: \(1.91178\times 10^{7}\)
Root analytic conductor: \(8.13167\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{8281} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(4\)
Selberg data: \((8,\ 7^{8} \cdot 13^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

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)$
bad7 \( 1 \)
13 \( 1 \)
good2$C_2$ \( ( 1 - T + p T^{2} )^{4} \)
3$C_2^2$ \( ( 1 + 4 T^{2} + p^{2} T^{4} )^{2} \)
5$D_4\times C_2$ \( 1 - 4 T^{2} + 31 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} \)
11$D_{4}$ \( ( 1 - 2 T - 2 p T^{3} + p^{2} T^{4} )^{2} \)
17$D_4\times C_2$ \( 1 + 36 T^{2} + 695 T^{4} + 36 p^{2} T^{6} + p^{4} T^{8} \)
19$C_2^2$ \( ( 1 + 30 T^{2} + p^{2} T^{4} )^{2} \)
23$D_{4}$ \( ( 1 + 6 T + 32 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \)
29$D_{4}$ \( ( 1 + 8 T + 51 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \)
31$C_2^2$ \( ( 1 + 60 T^{2} + p^{2} T^{4} )^{2} \)
37$D_{4}$ \( ( 1 + 4 T + 55 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \)
41$D_4\times C_2$ \( 1 + 60 T^{2} + 2399 T^{4} + 60 p^{2} T^{6} + p^{4} T^{8} \)
43$D_{4}$ \( ( 1 - 6 T + 72 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \)
47$C_2^2$ \( ( 1 + 86 T^{2} + p^{2} T^{4} )^{2} \)
53$D_{4}$ \( ( 1 + 6 T + 23 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \)
59$D_4\times C_2$ \( 1 + 80 T^{2} + 2674 T^{4} + 80 p^{2} T^{6} + p^{4} T^{8} \)
61$D_4\times C_2$ \( 1 + 172 T^{2} + 13711 T^{4} + 172 p^{2} T^{6} + p^{4} T^{8} \)
67$D_{4}$ \( ( 1 - 2 T + 112 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \)
71$C_2$ \( ( 1 - 6 T + p T^{2} )^{4} \)
73$D_4\times C_2$ \( 1 + 100 T^{2} + 127 p T^{4} + 100 p^{2} T^{6} + p^{4} T^{8} \)
79$D_{4}$ \( ( 1 + 14 T + 184 T^{2} + 14 p T^{3} + p^{2} T^{4} )^{2} \)
83$C_2^2$ \( ( 1 + 68 T^{2} + p^{2} T^{4} )^{2} \)
89$D_4\times C_2$ \( 1 - 16 T^{2} + 14434 T^{4} - 16 p^{2} T^{6} + p^{4} T^{8} \)
97$C_2^2$ \( ( 1 + 176 T^{2} + p^{2} T^{4} )^{2} \)
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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

−5.89684027665791751504363136249, −5.49794841782714958544543515038, −5.38387949042911495030573417009, −5.35818273727973063038988670571, −5.12381173945497185200715969103, −4.86858001278640850079162306416, −4.78551870415636199723275737793, −4.67744596040200875337859874821, −4.27252186713279193301829993121, −3.97572472596881026942848007382, −3.95060620832981821275479969695, −3.93739922237364483028673655459, −3.72581325276870932629292132936, −3.58309319328120271482630293546, −3.38127624419801894077834048120, −3.32558283982703256973298074961, −2.77653583219558343960127506953, −2.61097585469625325153175250221, −2.58735428831980365958663128867, −2.56152764397577830063541114858, −2.19595322878054087625606661067, −1.66724417693147768874489687637, −1.28233310016117794014915772548, −1.26674383787486553216215737711, −0.71557325516128885097842539126, 0, 0, 0, 0, 0.71557325516128885097842539126, 1.26674383787486553216215737711, 1.28233310016117794014915772548, 1.66724417693147768874489687637, 2.19595322878054087625606661067, 2.56152764397577830063541114858, 2.58735428831980365958663128867, 2.61097585469625325153175250221, 2.77653583219558343960127506953, 3.32558283982703256973298074961, 3.38127624419801894077834048120, 3.58309319328120271482630293546, 3.72581325276870932629292132936, 3.93739922237364483028673655459, 3.95060620832981821275479969695, 3.97572472596881026942848007382, 4.27252186713279193301829993121, 4.67744596040200875337859874821, 4.78551870415636199723275737793, 4.86858001278640850079162306416, 5.12381173945497185200715969103, 5.35818273727973063038988670571, 5.38387949042911495030573417009, 5.49794841782714958544543515038, 5.89684027665791751504363136249

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