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Properties

Label 8-75e4-1.1-c11e4-0-5
Degree $8$
Conductor $31640625$
Sign $1$
Analytic cond. $1.10272\times 10^{7}$
Root an. cond. $7.59116$
Motivic weight $11$
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  + 4.73e3·4-s − 1.18e5·9-s − 7.23e5·11-s + 9.18e6·16-s + 3.11e7·19-s + 1.40e8·29-s + 5.97e8·31-s − 5.58e8·36-s − 9.29e8·41-s − 3.42e9·44-s + 2.61e9·49-s + 1.09e10·59-s + 2.91e10·61-s + 8.06e8·64-s + 2.24e9·71-s + 1.47e11·76-s + 1.58e11·79-s + 1.04e10·81-s − 4.42e10·89-s + 8.54e10·99-s + 4.53e11·101-s − 2.92e11·109-s + 6.65e11·116-s − 4.67e11·121-s + 2.82e12·124-s + 127-s + 131-s + ⋯
L(s)  = 1  + 2.31·4-s − 2/3·9-s − 1.35·11-s + 2.18·16-s + 2.88·19-s + 1.27·29-s + 3.74·31-s − 1.54·36-s − 1.25·41-s − 3.12·44-s + 1.32·49-s + 1.99·59-s + 4.41·61-s + 0.0939·64-s + 0.147·71-s + 6.66·76-s + 5.79·79-s + 1/3·81-s − 0.839·89-s + 0.903·99-s + 4.29·101-s − 1.82·109-s + 2.94·116-s − 1.63·121-s + 8.65·124-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 31640625 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31640625 ^{s/2} \, \Gamma_{\C}(s+11/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(31640625\)    =    \(3^{4} \cdot 5^{8}\)
Sign: $1$
Analytic conductor: \(1.10272\times 10^{7}\)
Root analytic conductor: \(7.59116\)
Motivic weight: \(11\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 31640625,\ (\ :11/2, 11/2, 11/2, 11/2),\ 1)\)

Particular Values

\(L(6)\) \(\approx\) \(18.36615919\)
\(L(\frac12)\) \(\approx\) \(18.36615919\)
\(L(\frac{13}{2})\) 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$C_2$ \( ( 1 + p^{10} T^{2} )^{2} \)
5 \( 1 \)
good2$D_4\times C_2$ \( 1 - 1183 p^{2} T^{2} + 51593 p^{8} T^{4} - 1183 p^{24} T^{6} + p^{44} T^{8} \)
7$D_4\times C_2$ \( 1 - 2610823356 T^{2} + 9214543582009316518 T^{4} - 2610823356 p^{22} T^{6} + p^{44} T^{8} \)
11$D_{4}$ \( ( 1 + 361792 T + 429852327334 T^{2} + 361792 p^{11} T^{3} + p^{22} T^{4} )^{2} \)
13$D_4\times C_2$ \( 1 - 4880421994028 T^{2} + \)\(12\!\cdots\!38\)\( T^{4} - 4880421994028 p^{22} T^{6} + p^{44} T^{8} \)
17$D_4\times C_2$ \( 1 - 91952393456860 T^{2} + \)\(40\!\cdots\!78\)\( T^{4} - 91952393456860 p^{22} T^{6} + p^{44} T^{8} \)
19$D_{4}$ \( ( 1 - 15562224 T + 223670591350918 T^{2} - 15562224 p^{11} T^{3} + p^{22} T^{4} )^{2} \)
23$D_4\times C_2$ \( 1 - 68211714589828 p T^{2} + \)\(13\!\cdots\!18\)\( T^{4} - 68211714589828 p^{23} T^{6} + p^{44} T^{8} \)
29$D_{4}$ \( ( 1 - 70320668 T + 10276443921751438 T^{2} - 70320668 p^{11} T^{3} + p^{22} T^{4} )^{2} \)
31$D_{4}$ \( ( 1 - 298584872 T + 71113481278713662 T^{2} - 298584872 p^{11} T^{3} + p^{22} T^{4} )^{2} \)
37$D_4\times C_2$ \( 1 - 421595876055422156 T^{2} + \)\(10\!\cdots\!18\)\( T^{4} - 421595876055422156 p^{22} T^{6} + p^{44} T^{8} \)
41$D_{4}$ \( ( 1 + 464942588 T + 352557098230423222 T^{2} + 464942588 p^{11} T^{3} + p^{22} T^{4} )^{2} \)
43$D_4\times C_2$ \( 1 + 1739218761373966420 T^{2} + \)\(23\!\cdots\!98\)\( T^{4} + 1739218761373966420 p^{22} T^{6} + p^{44} T^{8} \)
47$D_4\times C_2$ \( 1 + 947554235717506820 T^{2} + \)\(36\!\cdots\!18\)\( T^{4} + 947554235717506820 p^{22} T^{6} + p^{44} T^{8} \)
53$D_4\times C_2$ \( 1 - 31853529349389345004 T^{2} + \)\(41\!\cdots\!58\)\( T^{4} - 31853529349389345004 p^{22} T^{6} + p^{44} T^{8} \)
59$D_{4}$ \( ( 1 - 5480385856 T + 33382416716089423558 T^{2} - 5480385856 p^{11} T^{3} + p^{22} T^{4} )^{2} \)
61$D_{4}$ \( ( 1 - 14557903980 T + \)\(13\!\cdots\!58\)\( T^{2} - 14557903980 p^{11} T^{3} + p^{22} T^{4} )^{2} \)
67$D_4\times C_2$ \( 1 - \)\(35\!\cdots\!60\)\( T^{2} + \)\(60\!\cdots\!78\)\( T^{4} - \)\(35\!\cdots\!60\)\( p^{22} T^{6} + p^{44} T^{8} \)
71$D_{4}$ \( ( 1 - 1120561024 T + \)\(32\!\cdots\!86\)\( T^{2} - 1120561024 p^{11} T^{3} + p^{22} T^{4} )^{2} \)
73$D_4\times C_2$ \( 1 - \)\(93\!\cdots\!04\)\( T^{2} + \)\(41\!\cdots\!58\)\( T^{4} - \)\(93\!\cdots\!04\)\( p^{22} T^{6} + p^{44} T^{8} \)
79$D_{4}$ \( ( 1 - 79243055560 T + \)\(30\!\cdots\!58\)\( T^{2} - 79243055560 p^{11} T^{3} + p^{22} T^{4} )^{2} \)
83$D_4\times C_2$ \( 1 + \)\(10\!\cdots\!12\)\( T^{2} + \)\(33\!\cdots\!38\)\( T^{4} + \)\(10\!\cdots\!12\)\( p^{22} T^{6} + p^{44} T^{8} \)
89$D_{4}$ \( ( 1 + 22117321236 T + \)\(55\!\cdots\!78\)\( T^{2} + 22117321236 p^{11} T^{3} + p^{22} T^{4} )^{2} \)
97$D_4\times C_2$ \( 1 - \)\(15\!\cdots\!00\)\( T^{2} + \)\(16\!\cdots\!18\)\( T^{4} - \)\(15\!\cdots\!00\)\( p^{22} T^{6} + p^{44} T^{8} \)
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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

−8.414497431405509659779535358698, −7.924889929870609965845269216955, −7.75888863040698437473995413328, −7.65227739341583084782875254275, −7.17203001795716676941158494298, −6.81222318746166494471202684050, −6.53786863253795345877608100394, −6.33205398226788456268626331126, −6.31779127817341486642002346060, −5.38044355627602331430272760612, −5.31789854003381094795853517070, −5.21031321566623369953676269417, −4.88758257537652987617672333173, −4.22401924337297201030106978449, −3.66239732075049075345185679253, −3.46320695208396779168024063452, −3.01992030141895680142102801156, −2.57936601159975350636196837568, −2.57583924315007892381891304134, −2.30371294351774538913210604359, −2.00386766684850705902875382672, −1.14494354517351323466305785977, −1.04717866841636523079577278614, −0.75638130046991638438962421864, −0.44942137932042965092661217140, 0.44942137932042965092661217140, 0.75638130046991638438962421864, 1.04717866841636523079577278614, 1.14494354517351323466305785977, 2.00386766684850705902875382672, 2.30371294351774538913210604359, 2.57583924315007892381891304134, 2.57936601159975350636196837568, 3.01992030141895680142102801156, 3.46320695208396779168024063452, 3.66239732075049075345185679253, 4.22401924337297201030106978449, 4.88758257537652987617672333173, 5.21031321566623369953676269417, 5.31789854003381094795853517070, 5.38044355627602331430272760612, 6.31779127817341486642002346060, 6.33205398226788456268626331126, 6.53786863253795345877608100394, 6.81222318746166494471202684050, 7.17203001795716676941158494298, 7.65227739341583084782875254275, 7.75888863040698437473995413328, 7.924889929870609965845269216955, 8.414497431405509659779535358698

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