Properties

Label 8-75e4-1.1-c9e4-0-8
Degree $8$
Conductor $31640625$
Sign $1$
Analytic cond. $2.22635\times 10^{6}$
Root an. cond. $6.21511$
Motivic weight $9$
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  + 768·4-s − 1.31e4·9-s + 4.84e4·11-s + 3.27e5·16-s + 2.69e6·19-s + 8.56e6·29-s + 2.35e6·31-s − 1.00e7·36-s − 5.77e7·41-s + 3.72e7·44-s + 1.45e8·49-s + 2.30e8·59-s − 4.62e8·61-s + 2.51e8·64-s + 2.23e8·71-s + 2.07e9·76-s + 6.68e8·79-s + 1.29e8·81-s + 4.93e8·89-s − 6.35e8·99-s + 4.20e9·101-s − 2.29e9·109-s + 6.57e9·116-s − 5.49e9·121-s + 1.80e9·124-s + 127-s + 131-s + ⋯
L(s)  = 1  + 3/2·4-s − 2/3·9-s + 0.997·11-s + 1.24·16-s + 4.74·19-s + 2.24·29-s + 0.457·31-s − 36-s − 3.19·41-s + 1.49·44-s + 3.61·49-s + 2.47·59-s − 4.28·61-s + 1.87·64-s + 1.04·71-s + 7.12·76-s + 1.93·79-s + 1/3·81-s + 0.833·89-s − 0.665·99-s + 4.01·101-s − 1.55·109-s + 3.37·116-s − 2.32·121-s + 0.686·124-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 31640625 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31640625 ^{s/2} \, \Gamma_{\C}(s+9/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(31640625\)    =    \(3^{4} \cdot 5^{8}\)
Sign: $1$
Analytic conductor: \(2.22635\times 10^{6}\)
Root analytic conductor: \(6.21511\)
Motivic weight: \(9\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 31640625,\ (\ :9/2, 9/2, 9/2, 9/2),\ 1)\)

Particular Values

\(L(5)\) \(\approx\) \(17.05810167\)
\(L(\frac12)\) \(\approx\) \(17.05810167\)
\(L(\frac{11}{2})\) 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)$
bad3$C_2$ \( ( 1 + p^{8} T^{2} )^{2} \)
5 \( 1 \)
good2$D_4\times C_2$ \( 1 - 3 p^{8} T^{2} + 4097 p^{6} T^{4} - 3 p^{26} T^{6} + p^{36} T^{8} \)
7$D_4\times C_2$ \( 1 - 2977066 p^{2} T^{2} + 3549173410203 p^{4} T^{4} - 2977066 p^{20} T^{6} + p^{36} T^{8} \)
11$D_{4}$ \( ( 1 - 24228 T + 3626299434 T^{2} - 24228 p^{9} T^{3} + p^{18} T^{4} )^{2} \)
13$D_4\times C_2$ \( 1 - 29080408322 T^{2} + \)\(39\!\cdots\!03\)\( T^{4} - 29080408322 p^{18} T^{6} + p^{36} T^{8} \)
17$D_4\times C_2$ \( 1 - 278138381340 T^{2} + \)\(46\!\cdots\!18\)\( T^{4} - 278138381340 p^{18} T^{6} + p^{36} T^{8} \)
19$D_{4}$ \( ( 1 - 1348846 T + 918842277063 T^{2} - 1348846 p^{9} T^{3} + p^{18} T^{4} )^{2} \)
23$D_4\times C_2$ \( 1 - 4479978344756 T^{2} + \)\(11\!\cdots\!18\)\( T^{4} - 4479978344756 p^{18} T^{6} + p^{36} T^{8} \)
29$D_{4}$ \( ( 1 - 4283172 T + 28723511132238 T^{2} - 4283172 p^{9} T^{3} + p^{18} T^{4} )^{2} \)
31$D_{4}$ \( ( 1 - 1176582 T + 14878569678167 T^{2} - 1176582 p^{9} T^{3} + p^{18} T^{4} )^{2} \)
37$D_4\times C_2$ \( 1 - 398510416365644 T^{2} + \)\(72\!\cdots\!18\)\( T^{4} - 398510416365644 p^{18} T^{6} + p^{36} T^{8} \)
41$D_{4}$ \( ( 1 + 28888488 T + 440689642094322 T^{2} + 28888488 p^{9} T^{3} + p^{18} T^{4} )^{2} \)
43$D_4\times C_2$ \( 1 + 154763256370630 T^{2} + \)\(32\!\cdots\!23\)\( T^{4} + 154763256370630 p^{18} T^{6} + p^{36} T^{8} \)
47$D_4\times C_2$ \( 1 - 1638401810447220 T^{2} + \)\(17\!\cdots\!78\)\( T^{4} - 1638401810447220 p^{18} T^{6} + p^{36} T^{8} \)
53$D_4\times C_2$ \( 1 - 12822710492727156 T^{2} + \)\(62\!\cdots\!18\)\( T^{4} - 12822710492727156 p^{18} T^{6} + p^{36} T^{8} \)
59$D_{4}$ \( ( 1 - 115150284 T + 5714647579373658 T^{2} - 115150284 p^{9} T^{3} + p^{18} T^{4} )^{2} \)
61$D_{4}$ \( ( 1 + 231494410 T + 36715609466998803 T^{2} + 231494410 p^{9} T^{3} + p^{18} T^{4} )^{2} \)
67$D_4\times C_2$ \( 1 - 55867918378987210 T^{2} + \)\(21\!\cdots\!43\)\( T^{4} - 55867918378987210 p^{18} T^{6} + p^{36} T^{8} \)
71$D_{4}$ \( ( 1 - 111792024 T + 66109010369692206 T^{2} - 111792024 p^{9} T^{3} + p^{18} T^{4} )^{2} \)
73$D_4\times C_2$ \( 1 - 195076860465949916 T^{2} + \)\(16\!\cdots\!78\)\( T^{4} - 195076860465949916 p^{18} T^{6} + p^{36} T^{8} \)
79$D_{4}$ \( ( 1 - 334446000 T + 245839953959811038 T^{2} - 334446000 p^{9} T^{3} + p^{18} T^{4} )^{2} \)
83$D_4\times C_2$ \( 1 - 279618148088057732 T^{2} + \)\(84\!\cdots\!98\)\( T^{4} - 279618148088057732 p^{18} T^{6} + p^{36} T^{8} \)
89$D_{4}$ \( ( 1 - 246671136 T + 514300088985932338 T^{2} - 246671136 p^{9} T^{3} + p^{18} T^{4} )^{2} \)
97$D_4\times C_2$ \( 1 - 2706412353927631010 T^{2} + \)\(29\!\cdots\!03\)\( T^{4} - 2706412353927631010 p^{18} T^{6} + p^{36} T^{8} \)
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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

−8.999219892358494522897540744549, −8.322076369034610676780638786884, −8.212539036977626106564065641683, −7.907245924886590742752328977917, −7.53719114367035158289166294953, −7.19831633553957129404072276420, −6.88683892612369946804780256930, −6.76135759513963656312697095612, −6.52511407654740151124240880275, −5.85624258174516376125005778772, −5.63841924922583473154133558972, −5.52653724126582325021713510561, −4.89296893008769765287205051411, −4.86113216953950214430021180257, −4.20380811674274001154954729632, −3.51751335604292203547335572822, −3.29045833596075512734568510402, −3.23150414865083986713892775276, −2.83191970794946461908477777607, −2.16281455827095777576963886649, −2.11815914444332965347052707175, −1.31692325634457144515342379131, −1.07794462211414346286938381994, −0.870098369303347932951265044865, −0.51042537692671486620986065303, 0.51042537692671486620986065303, 0.870098369303347932951265044865, 1.07794462211414346286938381994, 1.31692325634457144515342379131, 2.11815914444332965347052707175, 2.16281455827095777576963886649, 2.83191970794946461908477777607, 3.23150414865083986713892775276, 3.29045833596075512734568510402, 3.51751335604292203547335572822, 4.20380811674274001154954729632, 4.86113216953950214430021180257, 4.89296893008769765287205051411, 5.52653724126582325021713510561, 5.63841924922583473154133558972, 5.85624258174516376125005778772, 6.52511407654740151124240880275, 6.76135759513963656312697095612, 6.88683892612369946804780256930, 7.19831633553957129404072276420, 7.53719114367035158289166294953, 7.907245924886590742752328977917, 8.212539036977626106564065641683, 8.322076369034610676780638786884, 8.999219892358494522897540744549

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