Properties

Label 8-48e4-1.1-c14e4-0-1
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  + 3.27e3·3-s + 1.65e6·7-s + 4.28e6·9-s − 2.48e8·13-s + 1.25e9·19-s + 5.41e9·21-s + 1.17e10·25-s + 8.61e9·27-s − 4.85e10·31-s − 2.29e11·37-s − 8.13e11·39-s + 4.75e11·43-s + 2.84e11·49-s + 4.10e12·57-s − 8.00e12·61-s + 7.09e12·63-s + 5.70e12·67-s + 2.81e13·73-s + 3.83e13·75-s + 8.02e13·79-s + 2.90e13·81-s − 4.10e14·91-s − 1.59e14·93-s + 1.78e14·97-s − 1.20e14·103-s + 6.29e14·109-s − 7.51e14·111-s + ⋯
L(s)  = 1  + 1.49·3-s + 2.00·7-s + 0.896·9-s − 3.95·13-s + 1.40·19-s + 3.00·21-s + 1.91·25-s + 0.823·27-s − 1.76·31-s − 2.41·37-s − 5.92·39-s + 1.74·43-s + 0.419·49-s + 2.09·57-s − 2.54·61-s + 1.80·63-s + 0.941·67-s + 2.54·73-s + 2.87·75-s + 4.18·79-s + 1.26·81-s − 7.94·91-s − 2.64·93-s + 2.20·97-s − 0.982·103-s + 3.44·109-s − 3.61·111-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

\(L(\frac{15}{2})\) \(\approx\) \(0.1020029470\)
\(L(\frac12)\) \(\approx\) \(0.1020029470\)
\(L(8)\) 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$C_2^2$ \( 1 - 364 p^{2} T + 982 p^{8} T^{2} - 364 p^{16} T^{3} + p^{28} T^{4} \)
good5$D_4\times C_2$ \( 1 - 2340356564 p T^{2} + 3457673480203648854 p^{2} T^{4} - 2340356564 p^{29} T^{6} + p^{56} T^{8} \)
7$D_{4}$ \( ( 1 - 118148 p T + 126230410986 p T^{2} - 118148 p^{15} T^{3} + p^{28} T^{4} )^{2} \)
11$D_4\times C_2$ \( 1 - 197439346040740 T^{2} + \)\(13\!\cdots\!22\)\( p^{2} T^{4} - 197439346040740 p^{28} T^{6} + p^{56} T^{8} \)
13$D_{4}$ \( ( 1 + 9546620 p T + 69192541609062 p^{2} T^{2} + 9546620 p^{15} T^{3} + p^{28} T^{4} )^{2} \)
17$D_4\times C_2$ \( 1 - 589435666177378180 T^{2} + \)\(14\!\cdots\!02\)\( T^{4} - 589435666177378180 p^{28} T^{6} + p^{56} T^{8} \)
19$D_{4}$ \( ( 1 - 626033804 T + 438515193360666966 T^{2} - 626033804 p^{14} T^{3} + p^{28} T^{4} )^{2} \)
23$D_4\times C_2$ \( 1 + 199818496616545340 T^{2} - \)\(42\!\cdots\!42\)\( p^{2} T^{4} + 199818496616545340 p^{28} T^{6} + p^{56} T^{8} \)
29$D_4\times C_2$ \( 1 - \)\(93\!\cdots\!04\)\( T^{2} + \)\(39\!\cdots\!26\)\( T^{4} - \)\(93\!\cdots\!04\)\( p^{28} T^{6} + p^{56} T^{8} \)
31$D_{4}$ \( ( 1 + 24272242756 T + \)\(12\!\cdots\!26\)\( T^{2} + 24272242756 p^{14} T^{3} + p^{28} T^{4} )^{2} \)
37$D_{4}$ \( ( 1 + 114680549804 T + \)\(14\!\cdots\!62\)\( T^{2} + 114680549804 p^{14} T^{3} + p^{28} T^{4} )^{2} \)
41$D_4\times C_2$ \( 1 - \)\(12\!\cdots\!00\)\( T^{2} + \)\(67\!\cdots\!22\)\( T^{4} - \)\(12\!\cdots\!00\)\( p^{28} T^{6} + p^{56} T^{8} \)
43$D_{4}$ \( ( 1 - 237561908012 T + \)\(16\!\cdots\!54\)\( T^{2} - 237561908012 p^{14} T^{3} + p^{28} T^{4} )^{2} \)
47$D_4\times C_2$ \( 1 - \)\(28\!\cdots\!76\)\( T^{2} + \)\(14\!\cdots\!66\)\( T^{4} - \)\(28\!\cdots\!76\)\( p^{28} T^{6} + p^{56} T^{8} \)
53$D_4\times C_2$ \( 1 - \)\(36\!\cdots\!80\)\( T^{2} + \)\(70\!\cdots\!42\)\( T^{4} - \)\(36\!\cdots\!80\)\( p^{28} T^{6} + p^{56} T^{8} \)
59$D_4\times C_2$ \( 1 - \)\(23\!\cdots\!20\)\( T^{2} + \)\(20\!\cdots\!62\)\( T^{4} - \)\(23\!\cdots\!20\)\( p^{28} T^{6} + p^{56} T^{8} \)
61$D_{4}$ \( ( 1 + 4004253466892 T + \)\(97\!\cdots\!78\)\( T^{2} + 4004253466892 p^{14} T^{3} + p^{28} T^{4} )^{2} \)
67$D_{4}$ \( ( 1 - 2853249094604 T + \)\(38\!\cdots\!82\)\( T^{2} - 2853249094604 p^{14} T^{3} + p^{28} T^{4} )^{2} \)
71$D_4\times C_2$ \( 1 - \)\(28\!\cdots\!04\)\( T^{2} + \)\(34\!\cdots\!26\)\( T^{4} - \)\(28\!\cdots\!04\)\( p^{28} T^{6} + p^{56} T^{8} \)
73$D_{4}$ \( ( 1 - 14067899961700 T + \)\(28\!\cdots\!98\)\( T^{2} - 14067899961700 p^{14} T^{3} + p^{28} T^{4} )^{2} \)
79$D_{4}$ \( ( 1 - 40146026361596 T + \)\(11\!\cdots\!86\)\( T^{2} - 40146026361596 p^{14} T^{3} + p^{28} T^{4} )^{2} \)
83$D_4\times C_2$ \( 1 + \)\(63\!\cdots\!48\)\( T^{2} + \)\(24\!\cdots\!38\)\( T^{4} + \)\(63\!\cdots\!48\)\( p^{28} T^{6} + p^{56} T^{8} \)
89$D_4\times C_2$ \( 1 + \)\(10\!\cdots\!60\)\( T^{2} + \)\(75\!\cdots\!82\)\( T^{4} + \)\(10\!\cdots\!60\)\( p^{28} T^{6} + p^{56} T^{8} \)
97$D_{4}$ \( ( 1 - 89122083287428 T + \)\(13\!\cdots\!14\)\( T^{2} - 89122083287428 p^{14} T^{3} + p^{28} T^{4} )^{2} \)
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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

−8.798366379233026169769500972892, −8.110575682282648562823737257381, −7.88473157564074941460019877437, −7.78534986335477734556372600676, −7.53129085534911287216901245588, −7.14389622537943727706497508787, −7.00096841687576001252727234190, −6.59650828070209463981858531506, −5.98935599131898395425323610342, −5.29350888994453149676966119325, −5.12935480233071780626140001011, −4.93838948632324050874656509376, −4.74054701609473715814142045572, −4.63663539857786964462267094201, −3.71445519937938472435195522167, −3.43352873947078025560916924695, −3.29556503336947620523551560491, −2.67994362442572165464232631080, −2.31963073522684836680723707240, −2.12329261504762666589099768289, −2.12093366387537786559672451382, −1.35316507826818761887348519972, −1.14302549409781896233341863764, −0.68827554442594133330583904802, −0.02844800182699903447021355505, 0.02844800182699903447021355505, 0.68827554442594133330583904802, 1.14302549409781896233341863764, 1.35316507826818761887348519972, 2.12093366387537786559672451382, 2.12329261504762666589099768289, 2.31963073522684836680723707240, 2.67994362442572165464232631080, 3.29556503336947620523551560491, 3.43352873947078025560916924695, 3.71445519937938472435195522167, 4.63663539857786964462267094201, 4.74054701609473715814142045572, 4.93838948632324050874656509376, 5.12935480233071780626140001011, 5.29350888994453149676966119325, 5.98935599131898395425323610342, 6.59650828070209463981858531506, 7.00096841687576001252727234190, 7.14389622537943727706497508787, 7.53129085534911287216901245588, 7.78534986335477734556372600676, 7.88473157564074941460019877437, 8.110575682282648562823737257381, 8.798366379233026169769500972892

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