Properties

Label 8-48e4-1.1-c16e4-0-0
Degree $8$
Conductor $5308416$
Sign $1$
Analytic cond. $3.68554\times 10^{7}$
Root an. cond. $8.82699$
Motivic weight $16$
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  + 1.24e4·3-s + 1.05e6·7-s + 7.20e7·9-s − 2.74e8·13-s + 4.13e10·19-s + 1.30e10·21-s + 4.23e10·25-s + 4.08e11·27-s + 1.71e12·31-s + 5.25e12·37-s − 3.41e12·39-s − 1.89e13·43-s − 9.14e13·49-s + 5.13e14·57-s − 9.84e14·61-s + 7.56e13·63-s + 7.98e14·67-s − 2.12e15·73-s + 5.26e14·75-s + 1.94e15·79-s + 3.94e15·81-s − 2.88e14·91-s + 2.12e16·93-s − 3.18e16·97-s − 3.52e16·103-s + 6.53e16·111-s − 1.98e16·117-s + ⋯
L(s)  = 1  + 1.89·3-s + 0.182·7-s + 1.67·9-s − 0.336·13-s + 2.43·19-s + 0.344·21-s + 0.277·25-s + 1.44·27-s + 2.00·31-s + 1.49·37-s − 0.637·39-s − 1.62·43-s − 2.75·49-s + 4.60·57-s − 5.13·61-s + 0.305·63-s + 1.96·67-s − 2.63·73-s + 0.525·75-s + 1.28·79-s + 2.12·81-s − 0.0613·91-s + 3.80·93-s − 4.06·97-s − 2.78·103-s + 2.83·111-s − 0.564·117-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5308416 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(17-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5308416 ^{s/2} \, \Gamma_{\C}(s+8)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(5308416\)    =    \(2^{16} \cdot 3^{4}\)
Sign: $1$
Analytic conductor: \(3.68554\times 10^{7}\)
Root analytic conductor: \(8.82699\)
Motivic weight: \(16\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 5308416,\ (\ :8, 8, 8, 8),\ 1)\)

Particular Values

\(L(\frac{17}{2})\) \(\approx\) \(2.968626802\)
\(L(\frac12)\) \(\approx\) \(2.968626802\)
\(L(9)\) 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$D_{4}$ \( 1 - 460 p^{3} T + 112742 p^{6} T^{2} - 460 p^{19} T^{3} + p^{32} T^{4} \)
good5$C_2^2 \wr C_2$ \( 1 - 8471280596 p T^{2} - 8952186757218521874 p^{5} T^{4} - 8471280596 p^{33} T^{6} + p^{64} T^{8} \)
7$D_{4}$ \( ( 1 - 75020 p T + 134516880282 p^{3} T^{2} - 75020 p^{17} T^{3} + p^{32} T^{4} )^{2} \)
11$C_2^2 \wr C_2$ \( 1 - 12354555851874284 p T^{2} + \)\(69\!\cdots\!66\)\( p^{2} T^{4} - 12354555851874284 p^{33} T^{6} + p^{64} T^{8} \)
13$D_{4}$ \( ( 1 + 10571660 p T + 5498519503620054 p^{2} T^{2} + 10571660 p^{17} T^{3} + p^{32} T^{4} )^{2} \)
17$C_2^2 \wr C_2$ \( 1 - 66674679099167946244 T^{2} + \)\(56\!\cdots\!06\)\( T^{4} - 66674679099167946244 p^{32} T^{6} + p^{64} T^{8} \)
19$D_{4}$ \( ( 1 - 20674369892 T + \)\(68\!\cdots\!78\)\( T^{2} - 20674369892 p^{16} T^{3} + p^{32} T^{4} )^{2} \)
23$C_2^2 \wr C_2$ \( 1 + \)\(68\!\cdots\!76\)\( T^{2} + \)\(85\!\cdots\!86\)\( T^{4} + \)\(68\!\cdots\!76\)\( p^{32} T^{6} + p^{64} T^{8} \)
29$C_2^2 \wr C_2$ \( 1 - \)\(43\!\cdots\!04\)\( T^{2} + \)\(13\!\cdots\!86\)\( T^{4} - \)\(43\!\cdots\!04\)\( p^{32} T^{6} + p^{64} T^{8} \)
31$D_{4}$ \( ( 1 - 856168243892 T + \)\(14\!\cdots\!78\)\( T^{2} - 856168243892 p^{16} T^{3} + p^{32} T^{4} )^{2} \)
37$D_{4}$ \( ( 1 - 2629335341860 T + \)\(23\!\cdots\!38\)\( T^{2} - 2629335341860 p^{16} T^{3} + p^{32} T^{4} )^{2} \)
41$C_2^2 \wr C_2$ \( 1 - \)\(83\!\cdots\!84\)\( T^{2} + \)\(65\!\cdots\!26\)\( T^{4} - \)\(83\!\cdots\!84\)\( p^{32} T^{6} + p^{64} T^{8} \)
43$D_{4}$ \( ( 1 + 9485084864860 T + \)\(84\!\cdots\!86\)\( T^{2} + 9485084864860 p^{16} T^{3} + p^{32} T^{4} )^{2} \)
47$C_2^2 \wr C_2$ \( 1 - \)\(20\!\cdots\!64\)\( T^{2} + \)\(16\!\cdots\!06\)\( T^{4} - \)\(20\!\cdots\!64\)\( p^{32} T^{6} + p^{64} T^{8} \)
53$C_2^2 \wr C_2$ \( 1 - \)\(11\!\cdots\!64\)\( T^{2} + \)\(57\!\cdots\!06\)\( T^{4} - \)\(11\!\cdots\!64\)\( p^{32} T^{6} + p^{64} T^{8} \)
59$C_2^2 \wr C_2$ \( 1 - \)\(32\!\cdots\!44\)\( T^{2} + \)\(11\!\cdots\!46\)\( T^{4} - \)\(32\!\cdots\!44\)\( p^{32} T^{6} + p^{64} T^{8} \)
61$D_{4}$ \( ( 1 + 492098526528092 T + \)\(13\!\cdots\!38\)\( T^{2} + 492098526528092 p^{16} T^{3} + p^{32} T^{4} )^{2} \)
67$D_{4}$ \( ( 1 - 399005728500740 T + \)\(36\!\cdots\!86\)\( T^{2} - 399005728500740 p^{16} T^{3} + p^{32} T^{4} )^{2} \)
71$C_2^2 \wr C_2$ \( 1 - \)\(70\!\cdots\!64\)\( T^{2} + \)\(38\!\cdots\!86\)\( p T^{4} - \)\(70\!\cdots\!64\)\( p^{32} T^{6} + p^{64} T^{8} \)
73$D_{4}$ \( ( 1 + 1064074159757180 T + \)\(15\!\cdots\!18\)\( T^{2} + 1064074159757180 p^{16} T^{3} + p^{32} T^{4} )^{2} \)
79$D_{4}$ \( ( 1 - 974906968346036 T + \)\(21\!\cdots\!66\)\( T^{2} - 974906968346036 p^{16} T^{3} + p^{32} T^{4} )^{2} \)
83$C_2^2 \wr C_2$ \( 1 - \)\(14\!\cdots\!44\)\( T^{2} + \)\(99\!\cdots\!06\)\( T^{4} - \)\(14\!\cdots\!44\)\( p^{32} T^{6} + p^{64} T^{8} \)
89$C_2^2 \wr C_2$ \( 1 + \)\(11\!\cdots\!36\)\( T^{2} + \)\(46\!\cdots\!66\)\( T^{4} + \)\(11\!\cdots\!36\)\( p^{32} T^{6} + p^{64} T^{8} \)
97$D_{4}$ \( ( 1 + 15925984078821020 T + \)\(18\!\cdots\!06\)\( T^{2} + 15925984078821020 p^{16} T^{3} + p^{32} T^{4} )^{2} \)
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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

−8.203355690630332133416668444238, −8.122873840179564382015536276606, −7.975188672215642119036186150997, −7.35856064517381134777108901661, −7.11377377797015572237153236763, −7.01654829440295181199030527227, −6.35511056912849096510783656654, −6.05644198488043264270203390735, −5.97813830037152323128259716296, −5.08466326795719341545660950980, −5.04944914296884321032589527010, −4.76974820624427942017295243776, −4.39152275258250546208803015593, −3.99023540705179334733629183199, −3.57188882865170739254252690196, −3.10987808410153099523738171570, −2.97379535691563807735312129149, −2.76712179590367712817337791404, −2.71701338998741648285152307978, −1.90069360346571354627913531591, −1.59967305413743713567177177909, −1.45330731329884210131299022337, −1.02602422970390645491923643855, −0.71934767940657638642818680432, −0.12986362779444332568070526975, 0.12986362779444332568070526975, 0.71934767940657638642818680432, 1.02602422970390645491923643855, 1.45330731329884210131299022337, 1.59967305413743713567177177909, 1.90069360346571354627913531591, 2.71701338998741648285152307978, 2.76712179590367712817337791404, 2.97379535691563807735312129149, 3.10987808410153099523738171570, 3.57188882865170739254252690196, 3.99023540705179334733629183199, 4.39152275258250546208803015593, 4.76974820624427942017295243776, 5.04944914296884321032589527010, 5.08466326795719341545660950980, 5.97813830037152323128259716296, 6.05644198488043264270203390735, 6.35511056912849096510783656654, 7.01654829440295181199030527227, 7.11377377797015572237153236763, 7.35856064517381134777108901661, 7.975188672215642119036186150997, 8.122873840179564382015536276606, 8.203355690630332133416668444238

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