Properties

Label 8-210e4-1.1-c7e4-0-0
Degree $8$
Conductor $1944810000$
Sign $1$
Analytic cond. $1.85198\times 10^{7}$
Root an. cond. $8.09943$
Motivic weight $7$
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  − 16·2-s − 54·3-s + 64·4-s + 250·5-s + 864·6-s − 77·7-s + 1.02e3·8-s + 729·9-s − 4.00e3·10-s + 5.32e3·11-s − 3.45e3·12-s − 1.37e4·13-s + 1.23e3·14-s − 1.35e4·15-s − 1.63e4·16-s − 2.17e4·17-s − 1.16e4·18-s − 4.46e4·19-s + 1.60e4·20-s + 4.15e3·21-s − 8.52e4·22-s + 4.61e4·23-s − 5.52e4·24-s + 1.56e4·25-s + 2.19e5·26-s + 3.93e4·27-s − 4.92e3·28-s + ⋯
L(s)  = 1  − 1.41·2-s − 1.15·3-s + 1/2·4-s + 0.894·5-s + 1.63·6-s − 0.0848·7-s + 0.707·8-s + 1/3·9-s − 1.26·10-s + 1.20·11-s − 0.577·12-s − 1.73·13-s + 0.119·14-s − 1.03·15-s − 16-s − 1.07·17-s − 0.471·18-s − 1.49·19-s + 0.447·20-s + 0.0979·21-s − 1.70·22-s + 0.790·23-s − 0.816·24-s + 1/5·25-s + 2.44·26-s + 0.384·27-s − 0.0424·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(8-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+7/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(1.85198\times 10^{7}\)
Root analytic conductor: \(8.09943\)
Motivic weight: \(7\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{210} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 7^{4} ,\ ( \ : 7/2, 7/2, 7/2, 7/2 ),\ 1 )\)

Particular Values

\(L(4)\) \(\approx\) \(0.4328965164\)
\(L(\frac12)\) \(\approx\) \(0.4328965164\)
\(L(\frac{9}{2})\) 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)$
bad2$C_2$ \( ( 1 + p^{3} T + p^{6} T^{2} )^{2} \)
3$C_2$ \( ( 1 + p^{3} T + p^{6} T^{2} )^{2} \)
5$C_2$ \( ( 1 - p^{3} T + p^{6} T^{2} )^{2} \)
7$C_2^2$ \( 1 + 11 p T - 16686 p^{2} T^{2} + 11 p^{8} T^{3} + p^{14} T^{4} \)
good11$D_4\times C_2$ \( 1 - 5325 T + 8917079 T^{2} + 104028113700 T^{3} - 554463676176000 T^{4} + 104028113700 p^{7} T^{5} + 8917079 p^{14} T^{6} - 5325 p^{21} T^{7} + p^{28} T^{8} \)
13$D_{4}$ \( ( 1 + 6860 T + 30763125 T^{2} + 6860 p^{7} T^{3} + p^{14} T^{4} )^{2} \)
17$D_4\times C_2$ \( 1 + 21786 T - 245466430 T^{2} - 2191260280320 T^{3} + 155401787483141439 T^{4} - 2191260280320 p^{7} T^{5} - 245466430 p^{14} T^{6} + 21786 p^{21} T^{7} + p^{28} T^{8} \)
19$D_4\times C_2$ \( 1 + 44692 T + 155005759 T^{2} + 128490974836 p T^{3} + 2243546510130328 p^{2} T^{4} + 128490974836 p^{8} T^{5} + 155005759 p^{14} T^{6} + 44692 p^{21} T^{7} + p^{28} T^{8} \)
23$D_4\times C_2$ \( 1 - 46143 T - 5060519551 T^{2} - 17536421326158 T^{3} + 32849418093168952488 T^{4} - 17536421326158 p^{7} T^{5} - 5060519551 p^{14} T^{6} - 46143 p^{21} T^{7} + p^{28} T^{8} \)
29$D_{4}$ \( ( 1 - 240654 T + 46924664026 T^{2} - 240654 p^{7} T^{3} + p^{14} T^{4} )^{2} \)
31$D_4\times C_2$ \( 1 - 341927 T + 33133884097 T^{2} - 9832097553266270 T^{3} + \)\(30\!\cdots\!84\)\( T^{4} - 9832097553266270 p^{7} T^{5} + 33133884097 p^{14} T^{6} - 341927 p^{21} T^{7} + p^{28} T^{8} \)
37$D_4\times C_2$ \( 1 + 258856 T - 138288065633 T^{2} + 3994339660102168 T^{3} + \)\(26\!\cdots\!84\)\( T^{4} + 3994339660102168 p^{7} T^{5} - 138288065633 p^{14} T^{6} + 258856 p^{21} T^{7} + p^{28} T^{8} \)
41$D_{4}$ \( ( 1 + 320979 T + 390391764850 T^{2} + 320979 p^{7} T^{3} + p^{14} T^{4} )^{2} \)
43$D_{4}$ \( ( 1 + 990095 T + 710946547038 T^{2} + 990095 p^{7} T^{3} + p^{14} T^{4} )^{2} \)
47$D_4\times C_2$ \( 1 + 1722897 T + 1230732895613 T^{2} + 1248057862170194790 T^{3} + \)\(12\!\cdots\!80\)\( T^{4} + 1248057862170194790 p^{7} T^{5} + 1230732895613 p^{14} T^{6} + 1722897 p^{21} T^{7} + p^{28} T^{8} \)
53$D_4\times C_2$ \( 1 + 2994681 T + 4391722702919 T^{2} + 6669062677748535408 T^{3} + \)\(90\!\cdots\!38\)\( T^{4} + 6669062677748535408 p^{7} T^{5} + 4391722702919 p^{14} T^{6} + 2994681 p^{21} T^{7} + p^{28} T^{8} \)
59$D_4\times C_2$ \( 1 - 14550 p T + 610437743558 T^{2} + 70579202720626800 p T^{3} - \)\(80\!\cdots\!97\)\( T^{4} + 70579202720626800 p^{8} T^{5} + 610437743558 p^{14} T^{6} - 14550 p^{22} T^{7} + p^{28} T^{8} \)
61$D_4\times C_2$ \( 1 - 3920558 T + 5365393559962 T^{2} - 14584067235347242880 T^{3} + \)\(41\!\cdots\!19\)\( T^{4} - 14584067235347242880 p^{7} T^{5} + 5365393559962 p^{14} T^{6} - 3920558 p^{21} T^{7} + p^{28} T^{8} \)
67$D_4\times C_2$ \( 1 - 4109921 T + 666683047435 T^{2} - 16864421188932515360 T^{3} + \)\(12\!\cdots\!64\)\( T^{4} - 16864421188932515360 p^{7} T^{5} + 666683047435 p^{14} T^{6} - 4109921 p^{21} T^{7} + p^{28} T^{8} \)
71$D_{4}$ \( ( 1 + 6286110 T + 344627063426 p T^{2} + 6286110 p^{7} T^{3} + p^{14} T^{4} )^{2} \)
73$D_4\times C_2$ \( 1 + 4115065 T - 5172409247975 T^{2} + 46797194075350390 T^{3} + \)\(17\!\cdots\!66\)\( T^{4} + 46797194075350390 p^{7} T^{5} - 5172409247975 p^{14} T^{6} + 4115065 p^{21} T^{7} + p^{28} T^{8} \)
79$D_4\times C_2$ \( 1 - 1753115 T - 22151757036269 T^{2} + 23110699233244746760 T^{3} + \)\(20\!\cdots\!80\)\( T^{4} + 23110699233244746760 p^{7} T^{5} - 22151757036269 p^{14} T^{6} - 1753115 p^{21} T^{7} + p^{28} T^{8} \)
83$D_{4}$ \( ( 1 - 15542832 T + 114172035982006 T^{2} - 15542832 p^{7} T^{3} + p^{14} T^{4} )^{2} \)
89$D_4\times C_2$ \( 1 - 5883408 T - 55762019151694 T^{2} - 11259896007487780800 T^{3} + \)\(46\!\cdots\!67\)\( T^{4} - 11259896007487780800 p^{7} T^{5} - 55762019151694 p^{14} T^{6} - 5883408 p^{21} T^{7} + p^{28} T^{8} \)
97$D_{4}$ \( ( 1 + 10759946 T + 151301298759714 T^{2} + 10759946 p^{7} T^{3} + p^{14} T^{4} )^{2} \)
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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

−8.077718916332658570302074165578, −7.18177238079959494628062817020, −7.14776212368631944717541650539, −6.79353791568557025089590029954, −6.57685585401877903540421326860, −6.51991383298462256255925604024, −6.21544330151626586319390342979, −6.15890219186583834970957317050, −5.40905717134839489466483942528, −4.97860550410443354553937435099, −4.94778298385283654416789831490, −4.71973274934370835972365290070, −4.64489517391602314203676362062, −4.24002524823097139830493390363, −3.58464890625676372936482979372, −3.26758239754677200868612978508, −2.95398341007060903414444177766, −2.52792913786362986094220346272, −2.20513815488988655206522057914, −1.74348728013400391263188772235, −1.62945905567107324099756295514, −1.15200108588981229292989999508, −0.845507230003732514004737597581, −0.37355626197340142012459877382, −0.20732614257625002937590611530, 0.20732614257625002937590611530, 0.37355626197340142012459877382, 0.845507230003732514004737597581, 1.15200108588981229292989999508, 1.62945905567107324099756295514, 1.74348728013400391263188772235, 2.20513815488988655206522057914, 2.52792913786362986094220346272, 2.95398341007060903414444177766, 3.26758239754677200868612978508, 3.58464890625676372936482979372, 4.24002524823097139830493390363, 4.64489517391602314203676362062, 4.71973274934370835972365290070, 4.94778298385283654416789831490, 4.97860550410443354553937435099, 5.40905717134839489466483942528, 6.15890219186583834970957317050, 6.21544330151626586319390342979, 6.51991383298462256255925604024, 6.57685585401877903540421326860, 6.79353791568557025089590029954, 7.14776212368631944717541650539, 7.18177238079959494628062817020, 8.077718916332658570302074165578

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