Properties

Label 8-2450e4-1.1-c1e4-0-3
Degree $8$
Conductor $3.603\times 10^{13}$
Sign $1$
Analytic cond. $146478.$
Root an. cond. $4.42304$
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  − 4·2-s + 10·4-s − 20·8-s + 8·11-s + 35·16-s − 32·22-s + 8·23-s + 16·29-s − 56·32-s − 8·37-s + 80·44-s − 32·46-s − 32·53-s − 64·58-s + 84·64-s + 20·67-s − 8·71-s + 32·74-s + 32·79-s − 7·81-s − 160·88-s + 80·92-s + 128·106-s + 4·107-s + 72·109-s − 16·113-s + 160·116-s + ⋯
L(s)  = 1  − 2.82·2-s + 5·4-s − 7.07·8-s + 2.41·11-s + 35/4·16-s − 6.82·22-s + 1.66·23-s + 2.97·29-s − 9.89·32-s − 1.31·37-s + 12.0·44-s − 4.71·46-s − 4.39·53-s − 8.40·58-s + 21/2·64-s + 2.44·67-s − 0.949·71-s + 3.71·74-s + 3.60·79-s − 7/9·81-s − 17.0·88-s + 8.34·92-s + 12.4·106-s + 0.386·107-s + 6.89·109-s − 1.50·113-s + 14.8·116-s + ⋯

Functional equation

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

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 5^{8} \cdot 7^{8}\)
Sign: $1$
Analytic conductor: \(146478.\)
Root analytic conductor: \(4.42304\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{4} \cdot 5^{8} \cdot 7^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(1.634791495\)
\(L(\frac12)\) \(\approx\) \(1.634791495\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 + T )^{4} \)
5 \( 1 \)
7 \( 1 \)
good3$C_2^3$ \( 1 + 7 T^{4} + p^{4} T^{8} \)
11$D_{4}$ \( ( 1 - 4 T + 15 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \)
13$D_4\times C_2$ \( 1 + 4 T^{2} + 166 T^{4} + 4 p^{2} T^{6} + p^{4} T^{8} \)
17$D_4\times C_2$ \( 1 + 8 T^{2} + 55 T^{4} + 8 p^{2} T^{6} + p^{4} T^{8} \)
19$D_4\times C_2$ \( 1 + 64 T^{2} + 1735 T^{4} + 64 p^{2} T^{6} + p^{4} T^{8} \)
23$D_{4}$ \( ( 1 - 4 T + 6 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \)
29$D_{4}$ \( ( 1 - 8 T + 30 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \)
31$C_2^2$ \( ( 1 + 30 T^{2} + p^{2} T^{4} )^{2} \)
37$D_{4}$ \( ( 1 + 4 T + 34 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \)
41$D_4\times C_2$ \( 1 + 104 T^{2} + 5527 T^{4} + 104 p^{2} T^{6} + p^{4} T^{8} \)
43$C_2^2$ \( ( 1 + 42 T^{2} + p^{2} T^{4} )^{2} \)
47$D_4\times C_2$ \( 1 + 140 T^{2} + 9142 T^{4} + 140 p^{2} T^{6} + p^{4} T^{8} \)
53$C_2$ \( ( 1 + 8 T + p T^{2} )^{4} \)
59$D_4\times C_2$ \( 1 + 176 T^{2} + 14002 T^{4} + 176 p^{2} T^{6} + p^{4} T^{8} \)
61$D_4\times C_2$ \( 1 + 196 T^{2} + 16870 T^{4} + 196 p^{2} T^{6} + p^{4} T^{8} \)
67$D_{4}$ \( ( 1 - 10 T + 115 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \)
71$C_2$ \( ( 1 + 2 T + p T^{2} )^{4} \)
73$D_4\times C_2$ \( 1 + 232 T^{2} + 23575 T^{4} + 232 p^{2} T^{6} + p^{4} T^{8} \)
79$D_{4}$ \( ( 1 - 16 T + 178 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{2} \)
83$D_4\times C_2$ \( 1 + 96 T^{2} + 13607 T^{4} + 96 p^{2} T^{6} + p^{4} T^{8} \)
89$D_4\times C_2$ \( 1 + 248 T^{2} + 30327 T^{4} + 248 p^{2} T^{6} + p^{4} T^{8} \)
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

−6.65713544732406862553084447638, −6.26835048139795830053082871328, −6.04226251209691073786490935069, −5.94916263445630647821480927845, −5.94656844142486364049030107273, −5.17062324025722617668500792791, −5.09075440200538523055893951284, −5.06193563229420834349361056893, −4.68012738724509960551107730283, −4.37825588297373083122858656818, −4.37069519026931108964095032958, −3.77565506100005475186410293347, −3.69502851434695940863490467317, −3.17946647503704182998950795168, −3.15037232200679174544362056490, −3.13981258706124937170125490493, −2.87210621576124059994467586693, −2.11780432399617664532591229956, −2.05367618060061561174683409001, −2.04117898929986723485972763240, −1.50972432718404506063358356314, −1.25992640437677168354135482499, −0.966691295744663708276159977512, −0.75984095754994742949055754212, −0.43026665370124702441174195180, 0.43026665370124702441174195180, 0.75984095754994742949055754212, 0.966691295744663708276159977512, 1.25992640437677168354135482499, 1.50972432718404506063358356314, 2.04117898929986723485972763240, 2.05367618060061561174683409001, 2.11780432399617664532591229956, 2.87210621576124059994467586693, 3.13981258706124937170125490493, 3.15037232200679174544362056490, 3.17946647503704182998950795168, 3.69502851434695940863490467317, 3.77565506100005475186410293347, 4.37069519026931108964095032958, 4.37825588297373083122858656818, 4.68012738724509960551107730283, 5.06193563229420834349361056893, 5.09075440200538523055893951284, 5.17062324025722617668500792791, 5.94656844142486364049030107273, 5.94916263445630647821480927845, 6.04226251209691073786490935069, 6.26835048139795830053082871328, 6.65713544732406862553084447638

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