Properties

Label 8-48e4-1.1-c14e4-0-3
Degree $8$
Conductor $5308416$
Sign $1$
Analytic cond. $1.26839\times 10^{7}$
Root an. cond. $7.72514$
Motivic weight $14$
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  − 2.19e3·3-s − 8.25e5·7-s + 1.59e6·9-s + 2.01e8·13-s − 1.31e9·19-s + 1.81e9·21-s + 8.76e9·25-s − 6.93e9·27-s + 3.49e10·31-s + 5.55e10·37-s − 4.42e11·39-s + 3.23e11·43-s − 2.13e12·49-s + 2.88e12·57-s − 4.17e12·61-s − 1.31e12·63-s + 1.09e13·67-s − 4.46e13·73-s − 1.92e13·75-s − 4.12e13·79-s + 1.02e13·81-s − 1.66e14·91-s − 7.67e13·93-s + 7.05e13·97-s − 1.93e14·103-s − 3.00e14·109-s − 1.22e14·111-s + ⋯
L(s)  = 1  − 1.00·3-s − 1.00·7-s + 0.334·9-s + 3.21·13-s − 1.47·19-s + 1.00·21-s + 1.43·25-s − 0.662·27-s + 1.27·31-s + 0.585·37-s − 3.22·39-s + 1.19·43-s − 3.15·49-s + 1.47·57-s − 1.32·61-s − 0.335·63-s + 1.80·67-s − 4.04·73-s − 1.44·75-s − 2.14·79-s + 0.448·81-s − 3.22·91-s − 1.27·93-s + 0.872·97-s − 1.57·103-s − 1.64·109-s − 0.587·111-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5308416 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(15-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5308416 ^{s/2} \, \Gamma_{\C}(s+7)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(5308416\)    =    \(2^{16} \cdot 3^{4}\)
Sign: $1$
Analytic conductor: \(1.26839\times 10^{7}\)
Root analytic conductor: \(7.72514\)
Motivic weight: \(14\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 5308416,\ (\ :7, 7, 7, 7),\ 1)\)

Particular Values

\(L(\frac{15}{2})\) \(\approx\) \(0.6945543524\)
\(L(\frac12)\) \(\approx\) \(0.6945543524\)
\(L(8)\) 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 \( 1 \)
3$D_{4}$ \( 1 + 244 p^{2} T + 1474 p^{7} T^{2} + 244 p^{16} T^{3} + p^{28} T^{4} \)
good5$C_2^2 \wr C_2$ \( 1 - 1753157972 p T^{2} + 12993265388837262 p^{5} T^{4} - 1753157972 p^{29} T^{6} + p^{56} T^{8} \)
7$D_{4}$ \( ( 1 + 58972 p T + 189295846314 p T^{2} + 58972 p^{15} T^{3} + p^{28} T^{4} )^{2} \)
11$C_2^2 \wr C_2$ \( 1 - 10073573939524 p^{2} T^{2} + \)\(44\!\cdots\!26\)\( p^{4} T^{4} - 10073573939524 p^{30} T^{6} + p^{56} T^{8} \)
13$D_{4}$ \( ( 1 - 7757572 p T + 59987978953062 p^{2} T^{2} - 7757572 p^{15} T^{3} + p^{28} T^{4} )^{2} \)
17$C_2^2 \wr C_2$ \( 1 - 127136234878459780 T^{2} + \)\(29\!\cdots\!18\)\( T^{4} - 127136234878459780 p^{28} T^{6} + p^{56} T^{8} \)
19$D_{4}$ \( ( 1 + 657473620 T + 1701983228472310038 T^{2} + 657473620 p^{14} T^{3} + p^{28} T^{4} )^{2} \)
23$C_2^2 \wr C_2$ \( 1 - 32366964844766210500 T^{2} + \)\(48\!\cdots\!78\)\( T^{4} - 32366964844766210500 p^{28} T^{6} + p^{56} T^{8} \)
29$C_2^2 \wr C_2$ \( 1 - 47831478366103027684 T^{2} + \)\(17\!\cdots\!86\)\( T^{4} - 47831478366103027684 p^{28} T^{6} + p^{56} T^{8} \)
31$D_{4}$ \( ( 1 - 17485355036 T + \)\(10\!\cdots\!66\)\( T^{2} - 17485355036 p^{14} T^{3} + p^{28} T^{4} )^{2} \)
37$D_{4}$ \( ( 1 - 27788394964 T + \)\(14\!\cdots\!78\)\( T^{2} - 27788394964 p^{14} T^{3} + p^{28} T^{4} )^{2} \)
41$C_2^2 \wr C_2$ \( 1 - \)\(12\!\cdots\!84\)\( T^{2} + \)\(70\!\cdots\!06\)\( T^{4} - \)\(12\!\cdots\!84\)\( p^{28} T^{6} + p^{56} T^{8} \)
43$D_{4}$ \( ( 1 - 161839464524 T + \)\(12\!\cdots\!98\)\( T^{2} - 161839464524 p^{14} T^{3} + p^{28} T^{4} )^{2} \)
47$C_2^2 \wr C_2$ \( 1 - \)\(62\!\cdots\!00\)\( T^{2} + \)\(20\!\cdots\!38\)\( T^{4} - \)\(62\!\cdots\!00\)\( p^{28} T^{6} + p^{56} T^{8} \)
53$C_2^2 \wr C_2$ \( 1 - \)\(46\!\cdots\!00\)\( T^{2} + \)\(32\!\cdots\!82\)\( p^{2} T^{4} - \)\(46\!\cdots\!00\)\( p^{28} T^{6} + p^{56} T^{8} \)
59$C_2^2 \wr C_2$ \( 1 - \)\(12\!\cdots\!84\)\( T^{2} + \)\(11\!\cdots\!06\)\( T^{4} - \)\(12\!\cdots\!84\)\( p^{28} T^{6} + p^{56} T^{8} \)
61$D_{4}$ \( ( 1 + 2085820813196 T + \)\(10\!\cdots\!86\)\( T^{2} + 2085820813196 p^{14} T^{3} + p^{28} T^{4} )^{2} \)
67$D_{4}$ \( ( 1 - 5482119968876 T + \)\(80\!\cdots\!58\)\( T^{2} - 5482119968876 p^{14} T^{3} + p^{28} T^{4} )^{2} \)
71$C_2^2 \wr C_2$ \( 1 - \)\(10\!\cdots\!84\)\( T^{2} + \)\(83\!\cdots\!86\)\( T^{4} - \)\(10\!\cdots\!84\)\( p^{28} T^{6} + p^{56} T^{8} \)
73$D_{4}$ \( ( 1 + 22322065461404 T + \)\(36\!\cdots\!18\)\( T^{2} + 22322065461404 p^{14} T^{3} + p^{28} T^{4} )^{2} \)
79$D_{4}$ \( ( 1 + 20607721289380 T + \)\(36\!\cdots\!38\)\( T^{2} + 20607721289380 p^{14} T^{3} + p^{28} T^{4} )^{2} \)
83$C_2^2 \wr C_2$ \( 1 - \)\(24\!\cdots\!00\)\( T^{2} + \)\(26\!\cdots\!78\)\( T^{4} - \)\(24\!\cdots\!00\)\( p^{28} T^{6} + p^{56} T^{8} \)
89$C_2^2 \wr C_2$ \( 1 - \)\(68\!\cdots\!04\)\( T^{2} + \)\(19\!\cdots\!66\)\( T^{4} - \)\(68\!\cdots\!04\)\( p^{28} T^{6} + p^{56} T^{8} \)
97$D_{4}$ \( ( 1 - 35264990307844 T + \)\(84\!\cdots\!38\)\( T^{2} - 35264990307844 p^{14} T^{3} + p^{28} T^{4} )^{2} \)
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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

−8.741041882578488345610641755937, −8.279013914356413838323428416084, −8.164145836027558682444335852450, −7.80516493014106532801755720181, −6.93036476099823718846800675103, −6.86894741561302067246247198377, −6.85428909975722511729737752207, −6.04120585608209777244823741247, −5.91778265418976259687176669599, −5.89737315662136118132870263346, −5.87686296034407915597454502475, −4.69529824862350936472104629776, −4.60621522681380915572752246091, −4.59876498641192301333511074104, −3.85020287135897876003167001895, −3.52966589162345860737797618794, −3.38643499093317843956968956712, −2.82933949539077152241887332985, −2.64780603235711734460717366554, −1.97433037067175401693761852007, −1.47884937610816446298617579059, −1.23921135602345219119623033472, −1.07428502462252301887398290352, −0.48708708419775397452129186530, −0.14396224229671043903876033999, 0.14396224229671043903876033999, 0.48708708419775397452129186530, 1.07428502462252301887398290352, 1.23921135602345219119623033472, 1.47884937610816446298617579059, 1.97433037067175401693761852007, 2.64780603235711734460717366554, 2.82933949539077152241887332985, 3.38643499093317843956968956712, 3.52966589162345860737797618794, 3.85020287135897876003167001895, 4.59876498641192301333511074104, 4.60621522681380915572752246091, 4.69529824862350936472104629776, 5.87686296034407915597454502475, 5.89737315662136118132870263346, 5.91778265418976259687176669599, 6.04120585608209777244823741247, 6.85428909975722511729737752207, 6.86894741561302067246247198377, 6.93036476099823718846800675103, 7.80516493014106532801755720181, 8.164145836027558682444335852450, 8.279013914356413838323428416084, 8.741041882578488345610641755937

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