Properties

Label 8-114e4-1.1-c1e4-0-1
Degree $8$
Conductor $168896016$
Sign $1$
Analytic cond. $0.686637$
Root an. cond. $0.954093$
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  + 2·2-s + 2·3-s + 4-s + 4·6-s + 8·7-s − 2·8-s + 3·9-s + 2·12-s − 12·13-s + 16·14-s − 4·16-s − 12·17-s + 6·18-s − 2·19-s + 16·21-s − 4·24-s − 8·25-s − 24·26-s + 10·27-s + 8·28-s − 2·32-s − 24·34-s + 3·36-s − 4·38-s − 24·39-s − 6·41-s + 32·42-s + ⋯
L(s)  = 1  + 1.41·2-s + 1.15·3-s + 1/2·4-s + 1.63·6-s + 3.02·7-s − 0.707·8-s + 9-s + 0.577·12-s − 3.32·13-s + 4.27·14-s − 16-s − 2.91·17-s + 1.41·18-s − 0.458·19-s + 3.49·21-s − 0.816·24-s − 8/5·25-s − 4.70·26-s + 1.92·27-s + 1.51·28-s − 0.353·32-s − 4.11·34-s + 1/2·36-s − 0.648·38-s − 3.84·39-s − 0.937·41-s + 4.93·42-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 3^{4} \cdot 19^{4}\)
Sign: $1$
Analytic conductor: \(0.686637\)
Root analytic conductor: \(0.954093\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{114} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{4} \cdot 3^{4} \cdot 19^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(3.003342403\)
\(L(\frac12)\) \(\approx\) \(3.003342403\)
\(L(\frac{3}{2})\) 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$C_2$ \( ( 1 - T + T^{2} )^{2} \)
3$C_2^2$ \( 1 - 2 T + T^{2} - 2 p T^{3} + p^{2} T^{4} \)
19$C_2^2$ \( 1 + 2 T - 15 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
good5$C_2^3$ \( 1 + 8 T^{2} + 39 T^{4} + 8 p^{2} T^{6} + p^{4} T^{8} \)
7$D_{4}$ \( ( 1 - 4 T + 12 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \)
11$D_4\times C_2$ \( 1 - 34 T^{2} + 507 T^{4} - 34 p^{2} T^{6} + p^{4} T^{8} \)
13$C_2^2$ \( ( 1 + 6 T + 25 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \)
17$D_4\times C_2$ \( 1 + 12 T + 86 T^{2} + 456 T^{3} + 1971 T^{4} + 456 p T^{5} + 86 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \)
23$C_2^3$ \( 1 - 4 T^{2} - 513 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} \)
29$C_2^3$ \( 1 - 52 T^{2} + 1863 T^{4} - 52 p^{2} T^{6} + p^{4} T^{8} \)
31$C_2^2$ \( ( 1 - 44 T^{2} + p^{2} T^{4} )^{2} \)
37$D_4\times C_2$ \( 1 - 88 T^{2} + 3810 T^{4} - 88 p^{2} T^{6} + p^{4} T^{8} \)
41$C_2^2$ \( ( 1 + 3 T - 32 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \)
43$D_4\times C_2$ \( 1 - 8 T - 14 T^{2} + 64 T^{3} + 1483 T^{4} + 64 p T^{5} - 14 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
47$D_4\times C_2$ \( 1 - 12 T + 56 T^{2} - 96 T^{3} + 111 T^{4} - 96 p T^{5} + 56 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \)
53$D_4\times C_2$ \( 1 + 12 T + 26 T^{2} + 144 T^{3} + 3483 T^{4} + 144 p T^{5} + 26 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \)
59$D_4\times C_2$ \( 1 - 18 T + 131 T^{2} - 1350 T^{3} + 14652 T^{4} - 1350 p T^{5} + 131 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \)
61$D_4\times C_2$ \( 1 - 8 T - 68 T^{2} - 80 T^{3} + 9799 T^{4} - 80 p T^{5} - 68 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
67$D_4\times C_2$ \( 1 + 6 T + 131 T^{2} + 714 T^{3} + 10476 T^{4} + 714 p T^{5} + 131 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
71$C_2^2$ \( ( 1 - 6 T - 35 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \)
73$D_4\times C_2$ \( 1 - 2 T - 47 T^{2} + 190 T^{3} - 3020 T^{4} + 190 p T^{5} - 47 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
79$C_2^3$ \( 1 + 86 T^{2} + 1155 T^{4} + 86 p^{2} T^{6} + p^{4} T^{8} \)
83$C_2^2$$\times$$C_2^2$ \( ( 1 - 18 T + 241 T^{2} - 18 p T^{3} + p^{2} T^{4} )( 1 + 18 T + 241 T^{2} + 18 p T^{3} + p^{2} T^{4} ) \)
89$D_4\times C_2$ \( 1 - 24 T + 278 T^{2} - 2880 T^{3} + 29619 T^{4} - 2880 p T^{5} + 278 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} \)
97$D_4\times C_2$ \( 1 + 18 T + 257 T^{2} + 2682 T^{3} + 23268 T^{4} + 2682 p T^{5} + 257 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \)
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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

−10.04563987942332538864544266026, −9.594401128101041311197933591779, −9.443051614271687020926982601934, −9.206538024876112526840697440671, −8.886082651476942872306757100776, −8.519337754733934367291991047061, −8.119244971647420069877070792288, −8.038604944968059247643151268766, −7.975051088910496573396710006444, −7.37587048204865065347075386722, −7.00212381133974054533424786471, −6.93738025068783551799881966503, −6.62789165325942128680394582565, −5.91312693089997754829289866918, −5.57999378999719840774515171077, −5.05717147512744661904457551374, −4.88440965723369080757589943641, −4.74228246569414955471926197973, −4.44535338726630202289195031464, −4.14371224039697263635162358299, −3.82709755954716876771361966832, −2.95364971832215458448376476736, −2.32452839037930250275437655529, −2.23429147108818505649651132313, −1.97480694420883125076863906035, 1.97480694420883125076863906035, 2.23429147108818505649651132313, 2.32452839037930250275437655529, 2.95364971832215458448376476736, 3.82709755954716876771361966832, 4.14371224039697263635162358299, 4.44535338726630202289195031464, 4.74228246569414955471926197973, 4.88440965723369080757589943641, 5.05717147512744661904457551374, 5.57999378999719840774515171077, 5.91312693089997754829289866918, 6.62789165325942128680394582565, 6.93738025068783551799881966503, 7.00212381133974054533424786471, 7.37587048204865065347075386722, 7.975051088910496573396710006444, 8.038604944968059247643151268766, 8.119244971647420069877070792288, 8.519337754733934367291991047061, 8.886082651476942872306757100776, 9.206538024876112526840697440671, 9.443051614271687020926982601934, 9.594401128101041311197933591779, 10.04563987942332538864544266026

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