Properties

Label 8-210e4-1.1-c1e4-0-1
Degree $8$
Conductor $1944810000$
Sign $1$
Analytic cond. $7.90652$
Root an. cond. $1.29493$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 4-s − 4·5-s + 9-s + 10·11-s + 14·19-s − 4·20-s + 5·25-s + 12·31-s + 36-s − 36·41-s + 10·44-s − 4·45-s − 2·49-s − 40·55-s + 8·59-s + 4·61-s − 64-s − 8·71-s + 14·76-s − 28·79-s + 20·89-s − 56·95-s + 10·99-s + 5·100-s − 16·101-s − 36·109-s + 47·121-s + ⋯
L(s)  = 1  + 1/2·4-s − 1.78·5-s + 1/3·9-s + 3.01·11-s + 3.21·19-s − 0.894·20-s + 25-s + 2.15·31-s + 1/6·36-s − 5.62·41-s + 1.50·44-s − 0.596·45-s − 2/7·49-s − 5.39·55-s + 1.04·59-s + 0.512·61-s − 1/8·64-s − 0.949·71-s + 1.60·76-s − 3.15·79-s + 2.11·89-s − 5.74·95-s + 1.00·99-s + 1/2·100-s − 1.59·101-s − 3.44·109-s + 4.27·121-s + ⋯

Functional equation

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

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(7.90652\)
Root analytic conductor: \(1.29493\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{210} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 7^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(1.725835364\)
\(L(\frac12)\) \(\approx\) \(1.725835364\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 - T^{2} + T^{4} \)
3$C_2^2$ \( 1 - T^{2} + T^{4} \)
5$C_2^2$ \( 1 + 4 T + 11 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
7$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
good11$C_2^2$ \( ( 1 - 5 T + 14 T^{2} - 5 p T^{3} + p^{2} T^{4} )^{2} \)
13$C_2^2$ \( ( 1 - 25 T^{2} + p^{2} T^{4} )^{2} \)
17$C_2^2$$\times$$C_2^2$ \( ( 1 - 8 T + 47 T^{2} - 8 p T^{3} + p^{2} T^{4} )( 1 + 8 T + 47 T^{2} + 8 p T^{3} + p^{2} T^{4} ) \)
19$C_2$ \( ( 1 - 8 T + p T^{2} )^{2}( 1 + T + p T^{2} )^{2} \)
23$C_2^3$ \( 1 + 37 T^{2} + 840 T^{4} + 37 p^{2} T^{6} + p^{4} T^{8} \)
29$C_2$ \( ( 1 + p T^{2} )^{4} \)
31$C_2^2$ \( ( 1 - 6 T + 5 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \)
37$C_2^3$ \( 1 + 49 T^{2} + 1032 T^{4} + 49 p^{2} T^{6} + p^{4} T^{8} \)
41$C_2$ \( ( 1 + 9 T + p T^{2} )^{4} \)
43$C_2^2$ \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{2} \)
47$C_2^3$ \( 1 - 75 T^{2} + 3416 T^{4} - 75 p^{2} T^{6} + p^{4} T^{8} \)
53$C_2^3$ \( 1 + 105 T^{2} + 8216 T^{4} + 105 p^{2} T^{6} + p^{4} T^{8} \)
59$C_2^2$ \( ( 1 - 4 T - 43 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \)
61$C_2^2$ \( ( 1 - 2 T - 57 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \)
67$C_2^3$ \( 1 + 98 T^{2} + 5115 T^{4} + 98 p^{2} T^{6} + p^{4} T^{8} \)
71$C_2$ \( ( 1 + 2 T + p T^{2} )^{4} \)
73$C_2^3$ \( 1 + 130 T^{2} + 11571 T^{4} + 130 p^{2} T^{6} + p^{4} T^{8} \)
79$C_2^2$ \( ( 1 + 14 T + 117 T^{2} + 14 p T^{3} + p^{2} T^{4} )^{2} \)
83$C_2^2$ \( ( 1 - 66 T^{2} + p^{2} T^{4} )^{2} \)
89$C_2^2$ \( ( 1 - 10 T + 11 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \)
97$C_2$ \( ( 1 - 18 T + p T^{2} )^{2}( 1 + 18 T + p T^{2} )^{2} \)
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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

−9.179084383082505216020231608307, −8.720812962208541053074749127106, −8.359264956592400458377634175427, −8.313293496572333223869127736661, −7.940106893322321348302467614680, −7.896998875658963270868561172397, −7.19436015372326548269831519251, −7.15796884035051249280712395022, −6.83579701117063113614557987644, −6.74528461368143807122517886646, −6.67120912910708044868001964401, −6.05064931187087773186773629645, −5.74320376230786955381978504158, −5.26274379355380393494651105177, −5.20261531446482925378923494495, −4.57687865075359049495786459565, −4.32491177067244991324368399008, −4.16994536794863619644109203645, −3.59722019306561726796420886424, −3.40335185576002824914397478097, −3.20795553169858875821681464774, −2.87473553503323535748350655839, −1.73328792566260583794607486458, −1.53774337077871594567512854657, −0.919603030003091998088567393704, 0.919603030003091998088567393704, 1.53774337077871594567512854657, 1.73328792566260583794607486458, 2.87473553503323535748350655839, 3.20795553169858875821681464774, 3.40335185576002824914397478097, 3.59722019306561726796420886424, 4.16994536794863619644109203645, 4.32491177067244991324368399008, 4.57687865075359049495786459565, 5.20261531446482925378923494495, 5.26274379355380393494651105177, 5.74320376230786955381978504158, 6.05064931187087773186773629645, 6.67120912910708044868001964401, 6.74528461368143807122517886646, 6.83579701117063113614557987644, 7.15796884035051249280712395022, 7.19436015372326548269831519251, 7.896998875658963270868561172397, 7.940106893322321348302467614680, 8.313293496572333223869127736661, 8.359264956592400458377634175427, 8.720812962208541053074749127106, 9.179084383082505216020231608307

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