Properties

Label 8-1-1.1-c51e4-0-0
Degree $8$
Conductor $1$
Sign $1$
Analytic cond. $73638.5$
Root an. cond. $4.05871$
Motivic weight $51$
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  + 3.27e7·2-s + 4.03e11·3-s + 2.19e13·4-s + 1.21e18·5-s + 1.32e19·6-s + 6.56e21·7-s − 8.19e22·8-s − 1.68e24·9-s + 3.97e25·10-s + 3.52e26·11-s + 8.85e24·12-s + 3.07e28·13-s + 2.15e29·14-s + 4.90e29·15-s − 3.49e30·16-s + 4.81e31·17-s − 5.53e31·18-s + 8.17e32·19-s + 2.66e31·20-s + 2.65e33·21-s + 1.15e34·22-s − 5.44e34·23-s − 3.31e34·24-s − 4.40e35·25-s + 1.00e36·26-s + 2.20e35·27-s + 1.44e35·28-s + ⋯
L(s)  = 1  + 0.690·2-s + 0.275·3-s + 0.00973·4-s + 1.82·5-s + 0.189·6-s + 1.85·7-s − 0.767·8-s − 0.784·9-s + 1.25·10-s + 0.980·11-s + 0.00268·12-s + 1.20·13-s + 1.27·14-s + 0.501·15-s − 0.690·16-s + 2.02·17-s − 0.541·18-s + 2.01·19-s + 0.0177·20-s + 0.509·21-s + 0.676·22-s − 1.02·23-s − 0.211·24-s − 0.991·25-s + 0.833·26-s + 0.0696·27-s + 0.0180·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(52-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+51/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(1\)
Sign: $1$
Analytic conductor: \(73638.5\)
Root analytic conductor: \(4.05871\)
Motivic weight: \(51\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 1,\ (\ :51/2, 51/2, 51/2, 51/2),\ 1)\)

Particular Values

\(L(26)\) \(\approx\) \(18.68356050\)
\(L(\frac12)\) \(\approx\) \(18.68356050\)
\(L(\frac{53}{2})\) 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)$
good2$C_2 \wr S_4$ \( 1 - 4094505 p^{3} T + 128299190695 p^{13} T^{2} + 23025772401099165 p^{21} T^{3} - 5761999364104204611 p^{37} T^{4} + 23025772401099165 p^{72} T^{5} + 128299190695 p^{115} T^{6} - 4094505 p^{156} T^{7} + p^{204} T^{8} \)
3$C_2 \wr S_4$ \( 1 - 14957917520 p^{3} T + 94076672964345445060 p^{9} T^{2} - \)\(12\!\cdots\!60\)\( p^{17} T^{3} + \)\(12\!\cdots\!46\)\( p^{29} T^{4} - \)\(12\!\cdots\!60\)\( p^{68} T^{5} + 94076672964345445060 p^{111} T^{6} - 14957917520 p^{156} T^{7} + p^{204} T^{8} \)
5$C_2 \wr S_4$ \( 1 - 242822622593511576 p T + \)\(61\!\cdots\!84\)\( p^{5} T^{2} - \)\(14\!\cdots\!44\)\( p^{10} T^{3} + \)\(82\!\cdots\!26\)\( p^{16} T^{4} - \)\(14\!\cdots\!44\)\( p^{61} T^{5} + \)\(61\!\cdots\!84\)\( p^{107} T^{6} - 242822622593511576 p^{154} T^{7} + p^{204} T^{8} \)
7$C_2 \wr S_4$ \( 1 - \)\(13\!\cdots\!00\)\( p^{2} T + \)\(48\!\cdots\!00\)\( p^{6} T^{2} - \)\(78\!\cdots\!00\)\( p^{10} T^{3} + \)\(15\!\cdots\!02\)\( p^{14} T^{4} - \)\(78\!\cdots\!00\)\( p^{61} T^{5} + \)\(48\!\cdots\!00\)\( p^{108} T^{6} - \)\(13\!\cdots\!00\)\( p^{155} T^{7} + p^{204} T^{8} \)
11$C_2 \wr S_4$ \( 1 - \)\(35\!\cdots\!48\)\( T + \)\(25\!\cdots\!48\)\( p^{2} T^{2} - \)\(42\!\cdots\!96\)\( p^{5} T^{3} + \)\(20\!\cdots\!70\)\( p^{9} T^{4} - \)\(42\!\cdots\!96\)\( p^{56} T^{5} + \)\(25\!\cdots\!48\)\( p^{104} T^{6} - \)\(35\!\cdots\!48\)\( p^{153} T^{7} + p^{204} T^{8} \)
13$C_2 \wr S_4$ \( 1 - \)\(18\!\cdots\!20\)\( p^{2} T + \)\(13\!\cdots\!40\)\( p^{2} T^{2} - \)\(25\!\cdots\!80\)\( p^{3} T^{3} + \)\(43\!\cdots\!82\)\( p^{6} T^{4} - \)\(25\!\cdots\!80\)\( p^{54} T^{5} + \)\(13\!\cdots\!40\)\( p^{104} T^{6} - \)\(18\!\cdots\!20\)\( p^{155} T^{7} + p^{204} T^{8} \)
17$C_2 \wr S_4$ \( 1 - \)\(28\!\cdots\!60\)\( p T + \)\(89\!\cdots\!80\)\( p^{2} T^{2} - \)\(15\!\cdots\!20\)\( p^{3} T^{3} + \)\(27\!\cdots\!18\)\( p^{4} T^{4} - \)\(15\!\cdots\!20\)\( p^{54} T^{5} + \)\(89\!\cdots\!80\)\( p^{104} T^{6} - \)\(28\!\cdots\!60\)\( p^{154} T^{7} + p^{204} T^{8} \)
19$C_2 \wr S_4$ \( 1 - \)\(81\!\cdots\!80\)\( T + \)\(33\!\cdots\!04\)\( p T^{2} - \)\(49\!\cdots\!40\)\( p^{3} T^{3} + \)\(60\!\cdots\!34\)\( p^{5} T^{4} - \)\(49\!\cdots\!40\)\( p^{54} T^{5} + \)\(33\!\cdots\!04\)\( p^{103} T^{6} - \)\(81\!\cdots\!80\)\( p^{153} T^{7} + p^{204} T^{8} \)
23$C_2 \wr S_4$ \( 1 + \)\(23\!\cdots\!60\)\( p T + \)\(17\!\cdots\!60\)\( p^{2} T^{2} + \)\(15\!\cdots\!60\)\( p^{4} T^{3} + \)\(24\!\cdots\!22\)\( p^{6} T^{4} + \)\(15\!\cdots\!60\)\( p^{55} T^{5} + \)\(17\!\cdots\!60\)\( p^{104} T^{6} + \)\(23\!\cdots\!60\)\( p^{154} T^{7} + p^{204} T^{8} \)
29$C_2 \wr S_4$ \( 1 - \)\(24\!\cdots\!20\)\( T + \)\(10\!\cdots\!16\)\( T^{2} - \)\(54\!\cdots\!60\)\( p T^{3} + \)\(59\!\cdots\!06\)\( p^{2} T^{4} - \)\(54\!\cdots\!60\)\( p^{52} T^{5} + \)\(10\!\cdots\!16\)\( p^{102} T^{6} - \)\(24\!\cdots\!20\)\( p^{153} T^{7} + p^{204} T^{8} \)
31$C_2 \wr S_4$ \( 1 + \)\(74\!\cdots\!72\)\( T + \)\(12\!\cdots\!28\)\( p T^{2} + \)\(19\!\cdots\!84\)\( p^{2} T^{3} + \)\(20\!\cdots\!70\)\( p^{3} T^{4} + \)\(19\!\cdots\!84\)\( p^{53} T^{5} + \)\(12\!\cdots\!28\)\( p^{103} T^{6} + \)\(74\!\cdots\!72\)\( p^{153} T^{7} + p^{204} T^{8} \)
37$C_2 \wr S_4$ \( 1 - \)\(92\!\cdots\!60\)\( T + \)\(35\!\cdots\!80\)\( p T^{2} + \)\(42\!\cdots\!80\)\( p^{2} T^{3} - \)\(53\!\cdots\!54\)\( p^{3} T^{4} + \)\(42\!\cdots\!80\)\( p^{53} T^{5} + \)\(35\!\cdots\!80\)\( p^{103} T^{6} - \)\(92\!\cdots\!60\)\( p^{153} T^{7} + p^{204} T^{8} \)
41$C_2 \wr S_4$ \( 1 - \)\(14\!\cdots\!68\)\( T + \)\(71\!\cdots\!28\)\( p T^{2} - \)\(20\!\cdots\!36\)\( p^{2} T^{3} + \)\(10\!\cdots\!70\)\( p^{3} T^{4} - \)\(20\!\cdots\!36\)\( p^{53} T^{5} + \)\(71\!\cdots\!28\)\( p^{103} T^{6} - \)\(14\!\cdots\!68\)\( p^{153} T^{7} + p^{204} T^{8} \)
43$C_2 \wr S_4$ \( 1 + \)\(89\!\cdots\!00\)\( p T + \)\(30\!\cdots\!00\)\( p^{2} T^{2} + \)\(17\!\cdots\!00\)\( p^{3} T^{3} + \)\(40\!\cdots\!98\)\( p^{4} T^{4} + \)\(17\!\cdots\!00\)\( p^{54} T^{5} + \)\(30\!\cdots\!00\)\( p^{104} T^{6} + \)\(89\!\cdots\!00\)\( p^{154} T^{7} + p^{204} T^{8} \)
47$C_2 \wr S_4$ \( 1 - \)\(14\!\cdots\!40\)\( p T + \)\(27\!\cdots\!20\)\( p^{2} T^{2} - \)\(41\!\cdots\!80\)\( p^{3} T^{3} + \)\(33\!\cdots\!78\)\( p^{4} T^{4} - \)\(41\!\cdots\!80\)\( p^{54} T^{5} + \)\(27\!\cdots\!20\)\( p^{104} T^{6} - \)\(14\!\cdots\!40\)\( p^{154} T^{7} + p^{204} T^{8} \)
53$C_2 \wr S_4$ \( 1 + \)\(46\!\cdots\!60\)\( T + \)\(11\!\cdots\!80\)\( T^{2} - \)\(14\!\cdots\!80\)\( T^{3} + \)\(49\!\cdots\!18\)\( T^{4} - \)\(14\!\cdots\!80\)\( p^{51} T^{5} + \)\(11\!\cdots\!80\)\( p^{102} T^{6} + \)\(46\!\cdots\!60\)\( p^{153} T^{7} + p^{204} T^{8} \)
59$C_2 \wr S_4$ \( 1 - \)\(11\!\cdots\!40\)\( T + \)\(43\!\cdots\!36\)\( T^{2} - \)\(78\!\cdots\!80\)\( T^{3} + \)\(10\!\cdots\!86\)\( T^{4} - \)\(78\!\cdots\!80\)\( p^{51} T^{5} + \)\(43\!\cdots\!36\)\( p^{102} T^{6} - \)\(11\!\cdots\!40\)\( p^{153} T^{7} + p^{204} T^{8} \)
61$C_2 \wr S_4$ \( 1 - \)\(34\!\cdots\!48\)\( T + \)\(36\!\cdots\!08\)\( T^{2} - \)\(87\!\cdots\!96\)\( T^{3} + \)\(55\!\cdots\!70\)\( T^{4} - \)\(87\!\cdots\!96\)\( p^{51} T^{5} + \)\(36\!\cdots\!08\)\( p^{102} T^{6} - \)\(34\!\cdots\!48\)\( p^{153} T^{7} + p^{204} T^{8} \)
67$C_2 \wr S_4$ \( 1 - \)\(30\!\cdots\!20\)\( T + \)\(47\!\cdots\!20\)\( T^{2} - \)\(12\!\cdots\!60\)\( T^{3} + \)\(91\!\cdots\!78\)\( T^{4} - \)\(12\!\cdots\!60\)\( p^{51} T^{5} + \)\(47\!\cdots\!20\)\( p^{102} T^{6} - \)\(30\!\cdots\!20\)\( p^{153} T^{7} + p^{204} T^{8} \)
71$C_2 \wr S_4$ \( 1 - \)\(39\!\cdots\!88\)\( T + \)\(95\!\cdots\!88\)\( T^{2} - \)\(15\!\cdots\!36\)\( T^{3} + \)\(26\!\cdots\!70\)\( T^{4} - \)\(15\!\cdots\!36\)\( p^{51} T^{5} + \)\(95\!\cdots\!88\)\( p^{102} T^{6} - \)\(39\!\cdots\!88\)\( p^{153} T^{7} + p^{204} T^{8} \)
73$C_2 \wr S_4$ \( 1 - \)\(10\!\cdots\!20\)\( T + \)\(74\!\cdots\!40\)\( T^{2} - \)\(35\!\cdots\!40\)\( T^{3} + \)\(13\!\cdots\!58\)\( T^{4} - \)\(35\!\cdots\!40\)\( p^{51} T^{5} + \)\(74\!\cdots\!40\)\( p^{102} T^{6} - \)\(10\!\cdots\!20\)\( p^{153} T^{7} + p^{204} T^{8} \)
79$C_2 \wr S_4$ \( 1 - \)\(40\!\cdots\!20\)\( T + \)\(18\!\cdots\!16\)\( T^{2} - \)\(34\!\cdots\!40\)\( T^{3} + \)\(11\!\cdots\!46\)\( T^{4} - \)\(34\!\cdots\!40\)\( p^{51} T^{5} + \)\(18\!\cdots\!16\)\( p^{102} T^{6} - \)\(40\!\cdots\!20\)\( p^{153} T^{7} + p^{204} T^{8} \)
83$C_2 \wr S_4$ \( 1 - \)\(10\!\cdots\!60\)\( T + \)\(32\!\cdots\!20\)\( T^{2} - \)\(23\!\cdots\!20\)\( T^{3} + \)\(37\!\cdots\!78\)\( T^{4} - \)\(23\!\cdots\!20\)\( p^{51} T^{5} + \)\(32\!\cdots\!20\)\( p^{102} T^{6} - \)\(10\!\cdots\!60\)\( p^{153} T^{7} + p^{204} T^{8} \)
89$C_2 \wr S_4$ \( 1 + \)\(90\!\cdots\!40\)\( T + \)\(11\!\cdots\!56\)\( T^{2} + \)\(71\!\cdots\!80\)\( T^{3} + \)\(47\!\cdots\!26\)\( T^{4} + \)\(71\!\cdots\!80\)\( p^{51} T^{5} + \)\(11\!\cdots\!56\)\( p^{102} T^{6} + \)\(90\!\cdots\!40\)\( p^{153} T^{7} + p^{204} T^{8} \)
97$C_2 \wr S_4$ \( 1 + \)\(13\!\cdots\!20\)\( T + \)\(81\!\cdots\!80\)\( T^{2} + \)\(24\!\cdots\!60\)\( T^{3} + \)\(68\!\cdots\!18\)\( T^{4} + \)\(24\!\cdots\!60\)\( p^{51} T^{5} + \)\(81\!\cdots\!80\)\( p^{102} T^{6} + \)\(13\!\cdots\!20\)\( p^{153} T^{7} + p^{204} T^{8} \)
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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

−13.83094411907150913716270458144, −12.97643889751393502709158085272, −12.03795238952520272059363439468, −11.99400602044885273180751543862, −11.20306248452225429016226034819, −11.06266090259761044962942321159, −9.792279032448167874248641477831, −9.603932279762246098714017371305, −9.483691877095220456525194371351, −8.387233229994476295859626624605, −7.961279974284214418241080204082, −7.88468642974278430361491106397, −6.48736573311957801296620234338, −6.30812417909032547209546068873, −5.48941031099175255744586727946, −5.47892337175584191873993891140, −5.08956176803864243723829146291, −4.10763182976580858574694507598, −3.51688233082781277606046266650, −3.31909674112389602885775829792, −2.31336045986708990103866891156, −2.06773246701006938715445410458, −1.40440026423039880609375628767, −1.16938621946237143718274013481, −0.62068469340329910125506647824, 0.62068469340329910125506647824, 1.16938621946237143718274013481, 1.40440026423039880609375628767, 2.06773246701006938715445410458, 2.31336045986708990103866891156, 3.31909674112389602885775829792, 3.51688233082781277606046266650, 4.10763182976580858574694507598, 5.08956176803864243723829146291, 5.47892337175584191873993891140, 5.48941031099175255744586727946, 6.30812417909032547209546068873, 6.48736573311957801296620234338, 7.88468642974278430361491106397, 7.961279974284214418241080204082, 8.387233229994476295859626624605, 9.483691877095220456525194371351, 9.603932279762246098714017371305, 9.792279032448167874248641477831, 11.06266090259761044962942321159, 11.20306248452225429016226034819, 11.99400602044885273180751543862, 12.03795238952520272059363439468, 12.97643889751393502709158085272, 13.83094411907150913716270458144

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