Properties

Label 8-33e4-1.1-c3e4-0-0
Degree $8$
Conductor $1185921$
Sign $1$
Analytic cond. $14.3720$
Root an. cond. $1.39537$
Motivic weight $3$
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  + 10·2-s − 3·3-s + 48·4-s − 21·5-s − 30·6-s + 37·7-s + 160·8-s − 210·10-s + 41·11-s − 144·12-s − 77·13-s + 370·14-s + 63·15-s + 480·16-s + 192·17-s − 52·19-s − 1.00e3·20-s − 111·21-s + 410·22-s − 148·23-s − 480·24-s + 70·25-s − 770·26-s + 1.77e3·28-s + 414·29-s + 630·30-s − 198·31-s + ⋯
L(s)  = 1  + 3.53·2-s − 0.577·3-s + 6·4-s − 1.87·5-s − 2.04·6-s + 1.99·7-s + 7.07·8-s − 6.64·10-s + 1.12·11-s − 3.46·12-s − 1.64·13-s + 7.06·14-s + 1.08·15-s + 15/2·16-s + 2.73·17-s − 0.627·19-s − 11.2·20-s − 1.15·21-s + 3.97·22-s − 1.34·23-s − 4.08·24-s + 0.559·25-s − 5.80·26-s + 11.9·28-s + 2.65·29-s + 3.83·30-s − 1.14·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1185921 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1185921 ^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(1185921\)    =    \(3^{4} \cdot 11^{4}\)
Sign: $1$
Analytic conductor: \(14.3720\)
Root analytic conductor: \(1.39537\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 1185921,\ (\ :3/2, 3/2, 3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(8.257491990\)
\(L(\frac12)\) \(\approx\) \(8.257491990\)
\(L(\frac{5}{2})\) 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$C_4$ \( 1 + p T + p^{2} T^{2} + p^{3} T^{3} + p^{4} T^{4} \)
11$C_4$ \( 1 - 41 T + 171 p T^{2} - 41 p^{3} T^{3} + p^{6} T^{4} \)
good2$C_4\times C_2$ \( 1 - 5 p T + 13 p^{2} T^{2} - 25 p^{3} T^{3} + 39 p^{4} T^{4} - 25 p^{6} T^{5} + 13 p^{8} T^{6} - 5 p^{10} T^{7} + p^{12} T^{8} \)
5$C_2^2:C_4$ \( 1 + 21 T + 371 T^{2} + 5331 T^{3} + 75256 T^{4} + 5331 p^{3} T^{5} + 371 p^{6} T^{6} + 21 p^{9} T^{7} + p^{12} T^{8} \)
7$C_2^2:C_4$ \( 1 - 37 T + 276 T^{2} + 1717 p T^{3} - 398611 T^{4} + 1717 p^{4} T^{5} + 276 p^{6} T^{6} - 37 p^{9} T^{7} + p^{12} T^{8} \)
13$C_2^2:C_4$ \( 1 + 77 T + 102 T^{2} - 92675 T^{3} - 2336149 T^{4} - 92675 p^{3} T^{5} + 102 p^{6} T^{6} + 77 p^{9} T^{7} + p^{12} T^{8} \)
17$C_2^2:C_4$ \( 1 - 192 T + 9721 T^{2} + 336564 T^{3} - 58738571 T^{4} + 336564 p^{3} T^{5} + 9721 p^{6} T^{6} - 192 p^{9} T^{7} + p^{12} T^{8} \)
19$C_2^2:C_4$ \( 1 + 52 T + 4395 T^{2} + 634622 T^{3} + 83733539 T^{4} + 634622 p^{3} T^{5} + 4395 p^{6} T^{6} + 52 p^{9} T^{7} + p^{12} T^{8} \)
23$D_{4}$ \( ( 1 + 74 T + 23283 T^{2} + 74 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
29$C_2^2:C_4$ \( 1 - 414 T + 118207 T^{2} - 26000652 T^{3} + 4686132205 T^{4} - 26000652 p^{3} T^{5} + 118207 p^{6} T^{6} - 414 p^{9} T^{7} + p^{12} T^{8} \)
31$C_2^2:C_4$ \( 1 + 198 T - 7527 T^{2} + 1052546 T^{3} + 1096985415 T^{4} + 1052546 p^{3} T^{5} - 7527 p^{6} T^{6} + 198 p^{9} T^{7} + p^{12} T^{8} \)
37$C_2^2:C_4$ \( 1 - 201 T + 16698 T^{2} + 5740795 T^{3} - 1719990309 T^{4} + 5740795 p^{3} T^{5} + 16698 p^{6} T^{6} - 201 p^{9} T^{7} + p^{12} T^{8} \)
41$C_2^2:C_4$ \( 1 + 129 T - 62650 T^{2} - 148419 p T^{3} + 3934113919 T^{4} - 148419 p^{4} T^{5} - 62650 p^{6} T^{6} + 129 p^{9} T^{7} + p^{12} T^{8} \)
43$D_{4}$ \( ( 1 + 66 T + 151283 T^{2} + 66 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
47$C_2^2:C_4$ \( 1 + 35 T - 62723 T^{2} - 27766685 T^{3} + 7503450804 T^{4} - 27766685 p^{3} T^{5} - 62723 p^{6} T^{6} + 35 p^{9} T^{7} + p^{12} T^{8} \)
53$C_2^2:C_4$ \( 1 - 188 T - 43463 T^{2} - 48799240 T^{3} + 31190841441 T^{4} - 48799240 p^{3} T^{5} - 43463 p^{6} T^{6} - 188 p^{9} T^{7} + p^{12} T^{8} \)
59$C_2^2:C_4$ \( 1 + 1320 T + 594431 T^{2} + 124006410 T^{3} + 29587885771 T^{4} + 124006410 p^{3} T^{5} + 594431 p^{6} T^{6} + 1320 p^{9} T^{7} + p^{12} T^{8} \)
61$C_2^2:C_4$ \( 1 - 1275 T + 398429 T^{2} + 290507655 T^{3} - 280558765844 T^{4} + 290507655 p^{3} T^{5} + 398429 p^{6} T^{6} - 1275 p^{9} T^{7} + p^{12} T^{8} \)
67$D_{4}$ \( ( 1 - 75 T + 209531 T^{2} - 75 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
71$C_2^2:C_4$ \( 1 - 117 T + 251153 T^{2} - 117855189 T^{3} + 146557541980 T^{4} - 117855189 p^{3} T^{5} + 251153 p^{6} T^{6} - 117 p^{9} T^{7} + p^{12} T^{8} \)
73$C_2^2:C_4$ \( 1 + 982 T + 25167 T^{2} - 89899270 T^{3} + 52391431241 T^{4} - 89899270 p^{3} T^{5} + 25167 p^{6} T^{6} + 982 p^{9} T^{7} + p^{12} T^{8} \)
79$C_2^2:C_4$ \( 1 + 1469 T + 1436772 T^{2} + 1372338877 T^{3} + 1206137476805 T^{4} + 1372338877 p^{3} T^{5} + 1436772 p^{6} T^{6} + 1469 p^{9} T^{7} + p^{12} T^{8} \)
83$C_2^2:C_4$ \( 1 + 967 T + 281752 T^{2} + 542972135 T^{3} + 732624179601 T^{4} + 542972135 p^{3} T^{5} + 281752 p^{6} T^{6} + 967 p^{9} T^{7} + p^{12} T^{8} \)
89$D_{4}$ \( ( 1 + 2712 T + 3219029 T^{2} + 2712 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
97$C_2^2:C_4$ \( 1 - 2391 T + 2091113 T^{2} - 1694593545 T^{3} + 1933106673916 T^{4} - 1694593545 p^{3} T^{5} + 2091113 p^{6} T^{6} - 2391 p^{9} T^{7} + p^{12} T^{8} \)
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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

−12.15730332438268020242428464379, −12.03141292043350120342321824481, −11.62576974985343586017989264488, −11.49880939283497253221336658269, −11.33346874425822489471502713512, −10.70044460939170448402830645241, −10.12221506956354486898514808243, −10.11212943517045394040769804444, −9.630306165396462956165125023871, −8.466988271850730746134044889090, −8.424983302359743157101804638660, −7.992817190709516680826597435467, −7.62710040456143444465234293019, −7.20378303211091615818816883884, −7.07621973916399725730868043001, −6.08992759903448719105875703499, −5.80339618885088969845666332095, −5.28583349530515123773559522562, −5.14045627139769617922465362099, −4.39173826430367779602657207404, −4.31363659325680666640310738387, −4.18133559360769541514077159739, −3.51780201975491016753875326774, −2.83092266897241737045707387899, −1.47964404028776953856935953483, 1.47964404028776953856935953483, 2.83092266897241737045707387899, 3.51780201975491016753875326774, 4.18133559360769541514077159739, 4.31363659325680666640310738387, 4.39173826430367779602657207404, 5.14045627139769617922465362099, 5.28583349530515123773559522562, 5.80339618885088969845666332095, 6.08992759903448719105875703499, 7.07621973916399725730868043001, 7.20378303211091615818816883884, 7.62710040456143444465234293019, 7.992817190709516680826597435467, 8.424983302359743157101804638660, 8.466988271850730746134044889090, 9.630306165396462956165125023871, 10.11212943517045394040769804444, 10.12221506956354486898514808243, 10.70044460939170448402830645241, 11.33346874425822489471502713512, 11.49880939283497253221336658269, 11.62576974985343586017989264488, 12.03141292043350120342321824481, 12.15730332438268020242428464379

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