Properties

Label 8-48e4-1.1-c8e4-0-0
Degree $8$
Conductor $5308416$
Sign $1$
Analytic cond. $146203.$
Root an. cond. $4.42201$
Motivic weight $8$
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  − 264·5-s − 4.37e3·9-s + 1.46e4·13-s + 3.32e5·17-s + 5.55e5·25-s + 2.34e6·29-s + 4.31e6·37-s + 9.03e6·41-s + 1.15e6·45-s + 1.32e7·49-s − 4.21e7·53-s − 4.81e7·61-s − 3.86e6·65-s − 2.12e7·73-s + 1.43e7·81-s − 8.78e7·85-s − 1.26e7·89-s + 2.63e8·97-s − 6.43e7·101-s + 4.37e8·109-s − 1.11e8·113-s − 6.40e7·117-s + 6.03e8·121-s − 3.11e8·125-s + 127-s + 131-s + 137-s + ⋯
L(s)  = 1  − 0.422·5-s − 2/3·9-s + 0.512·13-s + 3.98·17-s + 1.42·25-s + 3.31·29-s + 2.30·37-s + 3.19·41-s + 0.281·45-s + 2.30·49-s − 5.34·53-s − 3.47·61-s − 0.216·65-s − 0.747·73-s + 1/3·81-s − 1.68·85-s − 0.202·89-s + 2.97·97-s − 0.618·101-s + 3.10·109-s − 0.682·113-s − 0.341·117-s + 2.81·121-s − 1.27·125-s − 1.39·145-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5308416 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5308416 ^{s/2} \, \Gamma_{\C}(s+4)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(5308416\)    =    \(2^{16} \cdot 3^{4}\)
Sign: $1$
Analytic conductor: \(146203.\)
Root analytic conductor: \(4.42201\)
Motivic weight: \(8\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 5308416,\ (\ :4, 4, 4, 4),\ 1)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(7.170292918\)
\(L(\frac12)\) \(\approx\) \(7.170292918\)
\(L(5)\) 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$C_2$ \( ( 1 + p^{7} T^{2} )^{2} \)
good5$D_{4}$ \( ( 1 + 132 T - 50354 p T^{2} + 132 p^{8} T^{3} + p^{16} T^{4} )^{2} \)
7$D_4\times C_2$ \( 1 - 13259044 T^{2} + 36717651462 p^{4} T^{4} - 13259044 p^{16} T^{6} + p^{32} T^{8} \)
11$D_4\times C_2$ \( 1 - 54853100 p T^{2} + 177212290163841222 T^{4} - 54853100 p^{17} T^{6} + p^{32} T^{8} \)
13$D_{4}$ \( ( 1 - 7316 T + 711204006 T^{2} - 7316 p^{8} T^{3} + p^{16} T^{4} )^{2} \)
17$D_{4}$ \( ( 1 - 166452 T + 16890408614 T^{2} - 166452 p^{8} T^{3} + p^{16} T^{4} )^{2} \)
19$D_4\times C_2$ \( 1 + 1001189684 p T^{2} + \)\(64\!\cdots\!62\)\( T^{4} + 1001189684 p^{17} T^{6} + p^{32} T^{8} \)
23$D_4\times C_2$ \( 1 - 182823998212 T^{2} + \)\(17\!\cdots\!22\)\( T^{4} - 182823998212 p^{16} T^{6} + p^{32} T^{8} \)
29$D_{4}$ \( ( 1 - 1171788 T + 1170231310502 T^{2} - 1171788 p^{8} T^{3} + p^{16} T^{4} )^{2} \)
31$D_4\times C_2$ \( 1 - 537822717220 T^{2} - \)\(40\!\cdots\!38\)\( T^{4} - 537822717220 p^{16} T^{6} + p^{32} T^{8} \)
37$D_{4}$ \( ( 1 - 2157892 T + 4692196769862 T^{2} - 2157892 p^{8} T^{3} + p^{16} T^{4} )^{2} \)
41$D_{4}$ \( ( 1 - 4517748 T + 20265362518694 T^{2} - 4517748 p^{8} T^{3} + p^{16} T^{4} )^{2} \)
43$D_4\times C_2$ \( 1 - 20498394519844 T^{2} + \)\(26\!\cdots\!02\)\( T^{4} - 20498394519844 p^{16} T^{6} + p^{32} T^{8} \)
47$D_4\times C_2$ \( 1 - 92364303273604 T^{2} + \)\(32\!\cdots\!62\)\( T^{4} - 92364303273604 p^{16} T^{6} + p^{32} T^{8} \)
53$D_{4}$ \( ( 1 + 21093300 T + 233197435266086 T^{2} + 21093300 p^{8} T^{3} + p^{16} T^{4} )^{2} \)
59$D_4\times C_2$ \( 1 - 508895856458020 T^{2} + \)\(10\!\cdots\!82\)\( T^{4} - 508895856458020 p^{16} T^{6} + p^{32} T^{8} \)
61$D_{4}$ \( ( 1 + 24074204 T + 494460041894022 T^{2} + 24074204 p^{8} T^{3} + p^{16} T^{4} )^{2} \)
67$D_4\times C_2$ \( 1 - 171738736146724 T^{2} + \)\(92\!\cdots\!02\)\( T^{4} - 171738736146724 p^{16} T^{6} + p^{32} T^{8} \)
71$D_4\times C_2$ \( 1 - 733335921043204 T^{2} + \)\(65\!\cdots\!82\)\( T^{4} - 733335921043204 p^{16} T^{6} + p^{32} T^{8} \)
73$D_{4}$ \( ( 1 + 10607740 T + 942802630465926 T^{2} + 10607740 p^{8} T^{3} + p^{16} T^{4} )^{2} \)
79$D_4\times C_2$ \( 1 - 1493877002302756 T^{2} + \)\(42\!\cdots\!90\)\( T^{4} - 1493877002302756 p^{16} T^{6} + p^{32} T^{8} \)
83$D_4\times C_2$ \( 1 - 870883615601188 T^{2} + \)\(73\!\cdots\!98\)\( T^{4} - 870883615601188 p^{16} T^{6} + p^{32} T^{8} \)
89$D_{4}$ \( ( 1 + 6337788 T + 6738404128068998 T^{2} + 6337788 p^{8} T^{3} + p^{16} T^{4} )^{2} \)
97$D_{4}$ \( ( 1 - 131576900 T + 16326465568075398 T^{2} - 131576900 p^{8} T^{3} + p^{16} T^{4} )^{2} \)
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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

−10.04138084592359043206978962358, −9.439475885614498879091676345182, −9.344502548070733934440283570655, −8.855077400715678482763704031034, −8.655488273495206604475835029838, −7.955576637442554500544988710394, −7.83252884069898375721398766827, −7.71590481882873833704085154563, −7.53676666170279420993369713879, −6.79758137194928541844273173034, −6.12901552782594675965751690396, −6.09316296515965367551517464232, −6.00466435818141848267708240429, −5.37913291570465507491059868500, −4.70007298741614143316970914214, −4.66787507895472621210964477443, −4.26293231391293710499151709306, −3.32642054303944437633794558853, −3.26510463481869745562304685298, −2.86478863227993444940147936704, −2.64540174212908391507192801942, −1.57548975795808686366570010016, −1.02825423984603466199324502254, −0.940850889095659095560219094853, −0.54801821088576392674407038783, 0.54801821088576392674407038783, 0.940850889095659095560219094853, 1.02825423984603466199324502254, 1.57548975795808686366570010016, 2.64540174212908391507192801942, 2.86478863227993444940147936704, 3.26510463481869745562304685298, 3.32642054303944437633794558853, 4.26293231391293710499151709306, 4.66787507895472621210964477443, 4.70007298741614143316970914214, 5.37913291570465507491059868500, 6.00466435818141848267708240429, 6.09316296515965367551517464232, 6.12901552782594675965751690396, 6.79758137194928541844273173034, 7.53676666170279420993369713879, 7.71590481882873833704085154563, 7.83252884069898375721398766827, 7.955576637442554500544988710394, 8.655488273495206604475835029838, 8.855077400715678482763704031034, 9.344502548070733934440283570655, 9.439475885614498879091676345182, 10.04138084592359043206978962358

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