Properties

Label 8-12e8-1.1-c1e4-0-3
Degree $8$
Conductor $429981696$
Sign $1$
Analytic cond. $1.74806$
Root an. cond. $1.07230$
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 + 2·4-s + 4·5-s + 6·7-s − 4·8-s − 3·9-s − 8·10-s + 8·11-s + 14·13-s − 12·14-s + 8·16-s − 16·17-s + 6·18-s − 6·19-s + 8·20-s − 16·22-s − 6·23-s − 4·25-s − 28·26-s + 12·28-s + 6·29-s − 8·31-s − 8·32-s + 32·34-s + 24·35-s − 6·36-s + 12·37-s + ⋯
L(s)  = 1  − 1.41·2-s + 4-s + 1.78·5-s + 2.26·7-s − 1.41·8-s − 9-s − 2.52·10-s + 2.41·11-s + 3.88·13-s − 3.20·14-s + 2·16-s − 3.88·17-s + 1.41·18-s − 1.37·19-s + 1.78·20-s − 3.41·22-s − 1.25·23-s − 4/5·25-s − 5.49·26-s + 2.26·28-s + 1.11·29-s − 1.43·31-s − 1.41·32-s + 5.48·34-s + 4.05·35-s − 36-s + 1.97·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{16} \cdot 3^{8}\)
Sign: $1$
Analytic conductor: \(1.74806\)
Root analytic conductor: \(1.07230\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{144} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{16} \cdot 3^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(1.071542142\)
\(L(\frac12)\) \(\approx\) \(1.071542142\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 + p T + p T^{2} + p^{2} T^{3} + p^{2} T^{4} \)
3$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \)
good5$D_4\times C_2$ \( 1 - 4 T + 4 p T^{2} - 52 T^{3} + 151 T^{4} - 52 p T^{5} + 4 p^{3} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
7$D_4\times C_2$ \( 1 - 6 T + 4 p T^{2} - 96 T^{3} + 291 T^{4} - 96 p T^{5} + 4 p^{3} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
11$D_4\times C_2$ \( 1 - 8 T + 41 T^{2} - 152 T^{3} + 532 T^{4} - 152 p T^{5} + 41 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
13$C_2$$\times$$C_2^2$ \( ( 1 - 7 T + p T^{2} )^{2}( 1 - T^{2} + p^{2} T^{4} ) \)
17$C_2^2$ \( ( 1 + 8 T + 47 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \)
19$D_4\times C_2$ \( 1 + 6 T + 18 T^{2} + 132 T^{3} + 959 T^{4} + 132 p T^{5} + 18 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
23$D_4\times C_2$ \( 1 + 6 T + 52 T^{2} + 240 T^{3} + 1347 T^{4} + 240 p T^{5} + 52 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
29$C_2^3$ \( 1 - 6 T + 18 T^{2} - 36 T^{3} - 457 T^{4} - 36 p T^{5} + 18 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
31$D_4\times C_2$ \( 1 + 8 T - 2 T^{2} + 32 T^{3} + 1411 T^{4} + 32 p T^{5} - 2 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
37$D_4\times C_2$ \( 1 - 12 T + 72 T^{2} - 588 T^{3} + 4658 T^{4} - 588 p T^{5} + 72 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \)
41$C_2^3$ \( 1 + 73 T^{2} + 3648 T^{4} + 73 p^{2} T^{6} + p^{4} T^{8} \)
43$D_4\times C_2$ \( 1 - 2 T + 65 T^{2} - 426 T^{3} + 2744 T^{4} - 426 p T^{5} + 65 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
47$D_4\times C_2$ \( 1 + 2 T - 16 T^{2} - 148 T^{3} - 1997 T^{4} - 148 p T^{5} - 16 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
53$D_4\times C_2$ \( 1 + 16 T + 128 T^{2} + 976 T^{3} + 7378 T^{4} + 976 p T^{5} + 128 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \)
59$D_4\times C_2$ \( 1 - 12 T + 45 T^{2} + 828 T^{3} - 9748 T^{4} + 828 p T^{5} + 45 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \)
61$D_4\times C_2$ \( 1 - 24 T + 180 T^{2} + 300 T^{3} - 11209 T^{4} + 300 p T^{5} + 180 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} \)
67$D_4\times C_2$ \( 1 + 14 T + 113 T^{2} + 726 T^{3} + 3848 T^{4} + 726 p T^{5} + 113 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \)
71$D_4\times C_2$ \( 1 - 156 T^{2} + 13094 T^{4} - 156 p^{2} T^{6} + p^{4} T^{8} \)
73$D_4\times C_2$ \( 1 - 158 T^{2} + 16131 T^{4} - 158 p^{2} T^{6} + p^{4} T^{8} \)
79$C_2^2$ \( ( 1 + 12 T + 65 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \)
83$C_2^3$ \( 1 + 2 T + 2 T^{2} - 328 T^{3} - 7217 T^{4} - 328 p T^{5} + 2 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
89$C_2^2$ \( ( 1 - 174 T^{2} + p^{2} T^{4} )^{2} \)
97$D_4\times C_2$ \( 1 + 20 T + 109 T^{2} + 20 p T^{3} + 376 p T^{4} + 20 p^{2} T^{5} + 109 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} \)
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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

−9.690240391310992884490074308370, −8.949463079538421767894310194195, −8.940123209963376357906373123593, −8.925814671999008289510159098312, −8.690942904981951103212933914153, −8.451266136949706590916681764442, −8.317960413009994173860736631235, −7.86991057058215315590972851660, −7.68547276714428948630492265150, −6.81034816487123612174399055683, −6.67459875878768755936555091578, −6.37797699913261667604710731909, −6.15734370009165966506239904039, −6.06638622518396496763135835197, −5.78395227319685843449626916722, −5.54086619136761879072610801135, −4.65353205644240147684521217007, −4.34546274203470849940350431922, −4.17803090641290544437846190947, −3.79695875238478591206756505681, −3.28754684796516277935594559354, −2.26569361244856301475568674860, −2.03927772605864394853899138606, −1.74576706969998869124035091338, −1.27186660900970486748029768107, 1.27186660900970486748029768107, 1.74576706969998869124035091338, 2.03927772605864394853899138606, 2.26569361244856301475568674860, 3.28754684796516277935594559354, 3.79695875238478591206756505681, 4.17803090641290544437846190947, 4.34546274203470849940350431922, 4.65353205644240147684521217007, 5.54086619136761879072610801135, 5.78395227319685843449626916722, 6.06638622518396496763135835197, 6.15734370009165966506239904039, 6.37797699913261667604710731909, 6.67459875878768755936555091578, 6.81034816487123612174399055683, 7.68547276714428948630492265150, 7.86991057058215315590972851660, 8.317960413009994173860736631235, 8.451266136949706590916681764442, 8.690942904981951103212933914153, 8.925814671999008289510159098312, 8.940123209963376357906373123593, 8.949463079538421767894310194195, 9.690240391310992884490074308370

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