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Properties

Label 8-48e4-1.1-c14e4-0-2
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  − 2.14e3·3-s − 1.70e6·7-s − 5.61e5·9-s − 9.35e7·13-s + 1.34e9·19-s + 3.67e9·21-s + 1.01e10·25-s + 2.04e9·27-s − 4.65e10·31-s + 3.00e11·37-s + 2.00e11·39-s − 2.46e11·43-s + 9.31e11·49-s − 2.89e12·57-s + 8.00e12·61-s + 9.59e11·63-s − 3.82e13·67-s + 1.27e13·73-s − 2.18e13·75-s + 2.98e13·79-s − 2.30e12·81-s + 1.59e14·91-s + 1.00e14·93-s − 3.01e14·97-s + 2.36e14·103-s − 4.90e14·109-s − 6.46e14·111-s + ⋯
L(s)  = 1  − 0.982·3-s − 2.07·7-s − 0.117·9-s − 1.49·13-s + 1.50·19-s + 2.03·21-s + 1.66·25-s + 0.195·27-s − 1.69·31-s + 3.16·37-s + 1.46·39-s − 0.906·43-s + 1.37·49-s − 1.48·57-s + 2.54·61-s + 0.243·63-s − 6.31·67-s + 1.15·73-s − 1.63·75-s + 1.55·79-s − 0.100·81-s + 3.09·91-s + 1.66·93-s − 3.73·97-s + 1.92·103-s − 2.68·109-s − 3.11·111-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

\(L(\frac{15}{2})\) \(\approx\) \(0.02865855208\)
\(L(\frac12)\) \(\approx\) \(0.02865855208\)
\(L(8)\) 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$D_{4}$ \( 1 + 716 p T + 21298 p^{5} T^{2} + 716 p^{15} T^{3} + p^{28} T^{4} \)
good5$C_2^2 \wr C_2$ \( 1 - 407405956 p^{2} T^{2} + 759152286064671582 p^{3} T^{4} - 407405956 p^{30} T^{6} + p^{56} T^{8} \)
7$D_{4}$ \( ( 1 + 122092 p T + 89956519626 p T^{2} + 122092 p^{15} T^{3} + p^{28} T^{4} )^{2} \)
11$C_2^2 \wr C_2$ \( 1 - 3393527575684 p^{2} T^{2} + \)\(22\!\cdots\!06\)\( p^{3} T^{4} - 3393527575684 p^{30} T^{6} + p^{56} T^{8} \)
13$D_{4}$ \( ( 1 + 3596348 p T + 49247613035238 p^{2} T^{2} + 3596348 p^{15} T^{3} + p^{28} T^{4} )^{2} \)
17$C_2^2 \wr C_2$ \( 1 + 26525338407439484 T^{2} + \)\(56\!\cdots\!46\)\( T^{4} + 26525338407439484 p^{28} T^{6} + p^{56} T^{8} \)
19$D_{4}$ \( ( 1 - 674600348 T + 466543652679214518 T^{2} - 674600348 p^{14} T^{3} + p^{28} T^{4} )^{2} \)
23$C_2^2 \wr C_2$ \( 1 - 12900104245999050436 T^{2} + \)\(30\!\cdots\!86\)\( T^{4} - 12900104245999050436 p^{28} T^{6} + p^{56} T^{8} \)
29$C_2^2 \wr C_2$ \( 1 - \)\(88\!\cdots\!24\)\( T^{2} + \)\(35\!\cdots\!66\)\( T^{4} - \)\(88\!\cdots\!24\)\( p^{28} T^{6} + p^{56} T^{8} \)
31$D_{4}$ \( ( 1 + 23292606868 T + \)\(11\!\cdots\!98\)\( T^{2} + 23292606868 p^{14} T^{3} + p^{28} T^{4} )^{2} \)
37$D_{4}$ \( ( 1 - 150436608532 T + \)\(23\!\cdots\!34\)\( T^{2} - 150436608532 p^{14} T^{3} + p^{28} T^{4} )^{2} \)
41$C_2^2 \wr C_2$ \( 1 - \)\(10\!\cdots\!44\)\( T^{2} + \)\(50\!\cdots\!26\)\( T^{4} - \)\(10\!\cdots\!44\)\( p^{28} T^{6} + p^{56} T^{8} \)
43$D_{4}$ \( ( 1 + 123224379076 T + \)\(14\!\cdots\!42\)\( T^{2} + 123224379076 p^{14} T^{3} + p^{28} T^{4} )^{2} \)
47$C_2^2 \wr C_2$ \( 1 + \)\(15\!\cdots\!24\)\( T^{2} - \)\(18\!\cdots\!34\)\( T^{4} + \)\(15\!\cdots\!24\)\( p^{28} T^{6} + p^{56} T^{8} \)
53$C_2^2 \wr C_2$ \( 1 - \)\(24\!\cdots\!76\)\( T^{2} + \)\(33\!\cdots\!66\)\( T^{4} - \)\(24\!\cdots\!76\)\( p^{28} T^{6} + p^{56} T^{8} \)
59$C_2^2 \wr C_2$ \( 1 - \)\(54\!\cdots\!44\)\( T^{2} + \)\(42\!\cdots\!26\)\( T^{4} - \)\(54\!\cdots\!44\)\( p^{28} T^{6} + p^{56} T^{8} \)
61$D_{4}$ \( ( 1 - 4004166132148 T + \)\(11\!\cdots\!58\)\( T^{2} - 4004166132148 p^{14} T^{3} + p^{28} T^{4} )^{2} \)
67$D_{4}$ \( ( 1 + 19147454106724 T + \)\(16\!\cdots\!02\)\( T^{2} + 19147454106724 p^{14} T^{3} + p^{28} T^{4} )^{2} \)
71$C_2^2 \wr C_2$ \( 1 - \)\(22\!\cdots\!24\)\( T^{2} + \)\(24\!\cdots\!66\)\( T^{4} - \)\(22\!\cdots\!24\)\( p^{28} T^{6} + p^{56} T^{8} \)
73$D_{4}$ \( ( 1 - 6360703346788 T + \)\(12\!\cdots\!54\)\( T^{2} - 6360703346788 p^{14} T^{3} + p^{28} T^{4} )^{2} \)
79$D_{4}$ \( ( 1 - 14901701665964 T + \)\(36\!\cdots\!86\)\( T^{2} - 14901701665964 p^{14} T^{3} + p^{28} T^{4} )^{2} \)
83$C_2^2 \wr C_2$ \( 1 - \)\(27\!\cdots\!16\)\( T^{2} + \)\(29\!\cdots\!46\)\( T^{4} - \)\(27\!\cdots\!16\)\( p^{28} T^{6} + p^{56} T^{8} \)
89$C_2^2 \wr C_2$ \( 1 - \)\(46\!\cdots\!64\)\( T^{2} + \)\(10\!\cdots\!86\)\( T^{4} - \)\(46\!\cdots\!64\)\( p^{28} T^{6} + p^{56} T^{8} \)
97$D_{4}$ \( ( 1 + 150812809565756 T + \)\(13\!\cdots\!22\)\( T^{2} + 150812809565756 p^{14} T^{3} + p^{28} T^{4} )^{2} \)
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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

−8.865335148013818878352185586233, −8.315140226969521623477873801205, −7.86871293317143064322605228006, −7.42308645475588429873493410480, −7.34622729615987898530206107262, −7.05239660812225589179304389154, −6.60625124879191287529464931686, −6.34693092485697029364716129842, −5.91389886356493841104439338146, −5.88112922526613388304002590638, −5.37204758771295809215313233751, −5.07298219857306966476435356628, −4.69912766443864648027236055093, −4.48997812911918874573007481756, −3.79658554943875333135532890611, −3.47544861769601379502379361582, −3.32975505555445363099871710511, −2.61105621697507691207866294664, −2.56609736881773643995062303742, −2.51303146587195898171614744688, −1.42945567478261706688287363626, −1.26607110175152420450285654325, −0.923120722504172644915468737878, −0.32125232718155468239905567513, −0.04690341116027891668016458656, 0.04690341116027891668016458656, 0.32125232718155468239905567513, 0.923120722504172644915468737878, 1.26607110175152420450285654325, 1.42945567478261706688287363626, 2.51303146587195898171614744688, 2.56609736881773643995062303742, 2.61105621697507691207866294664, 3.32975505555445363099871710511, 3.47544861769601379502379361582, 3.79658554943875333135532890611, 4.48997812911918874573007481756, 4.69912766443864648027236055093, 5.07298219857306966476435356628, 5.37204758771295809215313233751, 5.88112922526613388304002590638, 5.91389886356493841104439338146, 6.34693092485697029364716129842, 6.60625124879191287529464931686, 7.05239660812225589179304389154, 7.34622729615987898530206107262, 7.42308645475588429873493410480, 7.86871293317143064322605228006, 8.315140226969521623477873801205, 8.865335148013818878352185586233

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