Properties

Label 16-75e16-1.1-c1e8-0-6
Degree $16$
Conductor $1.002\times 10^{30}$
Sign $1$
Analytic cond. $1.65652\times 10^{13}$
Root an. cond. $6.70192$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $8$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 4-s − 10·7-s − 10·13-s − 16-s + 2·19-s + 10·28-s − 6·31-s − 50·37-s + 29·49-s + 10·52-s − 54·61-s − 10·67-s − 30·73-s − 2·76-s + 28·79-s + 100·91-s − 70·103-s − 12·109-s + 10·112-s − 28·121-s + 6·124-s + 127-s + 131-s − 20·133-s + 137-s + 139-s + 50·148-s + ⋯
L(s)  = 1  − 1/2·4-s − 3.77·7-s − 2.77·13-s − 1/4·16-s + 0.458·19-s + 1.88·28-s − 1.07·31-s − 8.21·37-s + 29/7·49-s + 1.38·52-s − 6.91·61-s − 1.22·67-s − 3.51·73-s − 0.229·76-s + 3.15·79-s + 10.4·91-s − 6.89·103-s − 1.14·109-s + 0.944·112-s − 2.54·121-s + 0.538·124-s + 0.0887·127-s + 0.0873·131-s − 1.73·133-s + 0.0854·137-s + 0.0848·139-s + 4.10·148-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 5^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 5^{32}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(3^{16} \cdot 5^{32}\)
Sign: $1$
Analytic conductor: \(1.65652\times 10^{13}\)
Root analytic conductor: \(6.70192\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{5625} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(8\)
Selberg data: \((16,\ 3^{16} \cdot 5^{32} ,\ ( \ : [1/2]^{8} ),\ 1 )\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
good2 \( 1 + T^{2} + p T^{4} + 3 T^{6} + 25 T^{8} + 3 p^{2} T^{10} + p^{5} T^{12} + p^{6} T^{14} + p^{8} T^{16} \)
7 \( ( 1 + 5 T + 23 T^{2} + 75 T^{3} + 244 T^{4} + 75 p T^{5} + 23 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
11 \( 1 + 28 T^{2} + 503 T^{4} + 5436 T^{6} + 61120 T^{8} + 5436 p^{2} T^{10} + 503 p^{4} T^{12} + 28 p^{6} T^{14} + p^{8} T^{16} \)
13 \( ( 1 + 5 T + 42 T^{2} + 160 T^{3} + 749 T^{4} + 160 p T^{5} + 42 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
17 \( 1 + 36 T^{2} + 1007 T^{4} + 19188 T^{6} + 344160 T^{8} + 19188 p^{2} T^{10} + 1007 p^{4} T^{12} + 36 p^{6} T^{14} + p^{8} T^{16} \)
19 \( ( 1 - T + 62 T^{2} - 48 T^{3} + 1675 T^{4} - 48 p T^{5} + 62 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} )^{2} \)
23 \( 1 + 19 T^{2} + 997 T^{4} + 14497 T^{6} + 447820 T^{8} + 14497 p^{2} T^{10} + 997 p^{4} T^{12} + 19 p^{6} T^{14} + p^{8} T^{16} \)
29 \( 1 + 97 T^{2} + 4733 T^{4} + 181059 T^{6} + 5875420 T^{8} + 181059 p^{2} T^{10} + 4733 p^{4} T^{12} + 97 p^{6} T^{14} + p^{8} T^{16} \)
31 \( ( 1 + 3 T + 108 T^{2} + 266 T^{3} + 4815 T^{4} + 266 p T^{5} + 108 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
37 \( ( 1 + 25 T + 363 T^{2} + 3495 T^{3} + 24844 T^{4} + 3495 p T^{5} + 363 p^{2} T^{6} + 25 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
41 \( 1 + 43 T^{2} + 1133 T^{4} + 60501 T^{6} + 1445620 T^{8} + 60501 p^{2} T^{10} + 1133 p^{4} T^{12} + 43 p^{6} T^{14} + p^{8} T^{16} \)
43 \( ( 1 + 97 T^{2} + 5389 T^{4} + 97 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
47 \( 1 + 276 T^{2} + 35732 T^{4} + 2879148 T^{6} + 160338390 T^{8} + 2879148 p^{2} T^{10} + 35732 p^{4} T^{12} + 276 p^{6} T^{14} + p^{8} T^{16} \)
53 \( 1 + 244 T^{2} + 25412 T^{4} + 1584492 T^{6} + 82074070 T^{8} + 1584492 p^{2} T^{10} + 25412 p^{4} T^{12} + 244 p^{6} T^{14} + p^{8} T^{16} \)
59 \( 1 + 172 T^{2} + 24743 T^{4} + 2088324 T^{6} + 150835120 T^{8} + 2088324 p^{2} T^{10} + 24743 p^{4} T^{12} + 172 p^{6} T^{14} + p^{8} T^{16} \)
61 \( ( 1 + 27 T + 498 T^{2} + 5924 T^{3} + 54645 T^{4} + 5924 p T^{5} + 498 p^{2} T^{6} + 27 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
67 \( ( 1 + 5 T + 108 T^{2} + 460 T^{3} + 8789 T^{4} + 460 p T^{5} + 108 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
71 \( 1 + 273 T^{2} + 41753 T^{4} + 4454691 T^{6} + 360816420 T^{8} + 4454691 p^{2} T^{10} + 41753 p^{4} T^{12} + 273 p^{6} T^{14} + p^{8} T^{16} \)
73 \( ( 1 + 15 T + 207 T^{2} + 1145 T^{3} + 11484 T^{4} + 1145 p T^{5} + 207 p^{2} T^{6} + 15 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
79 \( ( 1 - 14 T + 297 T^{2} - 2682 T^{3} + 33535 T^{4} - 2682 p T^{5} + 297 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
83 \( 1 + 244 T^{2} + 43127 T^{4} + 5300892 T^{6} + 498372400 T^{8} + 5300892 p^{2} T^{10} + 43127 p^{4} T^{12} + 244 p^{6} T^{14} + p^{8} T^{16} \)
89 \( 1 + 112 T^{2} + 18263 T^{4} + 2446764 T^{6} + 185830120 T^{8} + 2446764 p^{2} T^{10} + 18263 p^{4} T^{12} + 112 p^{6} T^{14} + p^{8} T^{16} \)
97 \( ( 1 + 343 T^{2} + 90 T^{3} + 47719 T^{4} + 90 p T^{5} + 343 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−3.63994989493359497135434513797, −3.52452572625373174301221815875, −3.49125378666243292548000194798, −3.29461112384443291771735334398, −3.23768436832519086944858537549, −3.18342331559412819860895697584, −3.18057824830046074389810054013, −2.99884104090010664696669841388, −2.98474425958620840376468932498, −2.86906874291236737455751030968, −2.78519159088471294911580580861, −2.70939383462823917588567568602, −2.46530884317195469737717973359, −2.19637742640192218530850196446, −2.09342409604575823549867468609, −2.04336281156556266223190143203, −1.99136196643060706463903585600, −1.94151306311571515090739530360, −1.77810869741048875388059538695, −1.51316137125109064892158598114, −1.31384229179971658256029510171, −1.28771562876727834514880912486, −1.21574438650371910596505522522, −1.01157731877838223240981330124, −0.936072746479354619713873049443, 0, 0, 0, 0, 0, 0, 0, 0, 0.936072746479354619713873049443, 1.01157731877838223240981330124, 1.21574438650371910596505522522, 1.28771562876727834514880912486, 1.31384229179971658256029510171, 1.51316137125109064892158598114, 1.77810869741048875388059538695, 1.94151306311571515090739530360, 1.99136196643060706463903585600, 2.04336281156556266223190143203, 2.09342409604575823549867468609, 2.19637742640192218530850196446, 2.46530884317195469737717973359, 2.70939383462823917588567568602, 2.78519159088471294911580580861, 2.86906874291236737455751030968, 2.98474425958620840376468932498, 2.99884104090010664696669841388, 3.18057824830046074389810054013, 3.18342331559412819860895697584, 3.23768436832519086944858537549, 3.29461112384443291771735334398, 3.49125378666243292548000194798, 3.52452572625373174301221815875, 3.63994989493359497135434513797

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

Plot not available for L-functions of degree greater than 10.