Properties

Label 16-3120e8-1.1-c1e8-0-3
Degree $16$
Conductor $8.979\times 10^{27}$
Sign $1$
Analytic cond. $1.48406\times 10^{11}$
Root an. cond. $4.99132$
Motivic weight $1$
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  − 2·5-s − 4·9-s − 2·11-s + 2·25-s − 16·29-s + 14·41-s + 8·45-s + 19·49-s + 4·55-s + 4·59-s + 22·61-s − 30·71-s − 2·79-s + 10·81-s − 18·89-s + 8·99-s + 4·101-s − 44·109-s − 45·121-s + 6·125-s + 127-s + 131-s + 137-s + 139-s + 32·145-s + 149-s + 151-s + ⋯
L(s)  = 1  − 0.894·5-s − 4/3·9-s − 0.603·11-s + 2/5·25-s − 2.97·29-s + 2.18·41-s + 1.19·45-s + 19/7·49-s + 0.539·55-s + 0.520·59-s + 2.81·61-s − 3.56·71-s − 0.225·79-s + 10/9·81-s − 1.90·89-s + 0.804·99-s + 0.398·101-s − 4.21·109-s − 4.09·121-s + 0.536·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 2.65·145-s + 0.0819·149-s + 0.0813·151-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8} \cdot 5^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8} \cdot 5^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{32} \cdot 3^{8} \cdot 5^{8} \cdot 13^{8}\)
Sign: $1$
Analytic conductor: \(1.48406\times 10^{11}\)
Root analytic conductor: \(4.99132\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{3120} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{32} \cdot 3^{8} \cdot 5^{8} \cdot 13^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.6019340333\)
\(L(\frac12)\) \(\approx\) \(0.6019340333\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( ( 1 + T^{2} )^{4} \)
5 \( 1 + 2 T + 2 T^{2} - 6 T^{3} - 46 T^{4} - 6 p T^{5} + 2 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
13 \( ( 1 + T^{2} )^{4} \)
good7 \( 1 - 19 T^{2} + 158 T^{4} - 765 T^{6} + 3714 T^{8} - 765 p^{2} T^{10} + 158 p^{4} T^{12} - 19 p^{6} T^{14} + p^{8} T^{16} \)
11 \( ( 1 + T + 24 T^{2} + 59 T^{3} + 282 T^{4} + 59 p T^{5} + 24 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} )^{2} \)
17 \( 1 - 75 T^{2} + 2778 T^{4} - 70725 T^{6} + 1368938 T^{8} - 70725 p^{2} T^{10} + 2778 p^{4} T^{12} - 75 p^{6} T^{14} + p^{8} T^{16} \)
19 \( ( 1 + p T^{2} )^{8} \)
23 \( 1 - 119 T^{2} + 6898 T^{4} - 258825 T^{6} + 6944954 T^{8} - 258825 p^{2} T^{10} + 6898 p^{4} T^{12} - 119 p^{6} T^{14} + p^{8} T^{16} \)
29 \( ( 1 + 8 T + 96 T^{2} + 552 T^{3} + 4142 T^{4} + 552 p T^{5} + 96 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
31 \( ( 1 + p T^{2} )^{8} \)
37 \( 1 - 163 T^{2} + 13874 T^{4} - 795269 T^{6} + 33768826 T^{8} - 795269 p^{2} T^{10} + 13874 p^{4} T^{12} - 163 p^{6} T^{14} + p^{8} T^{16} \)
41 \( ( 1 - 7 T + 156 T^{2} - 839 T^{3} + 9426 T^{4} - 839 p T^{5} + 156 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
43 \( 1 - 240 T^{2} + 28156 T^{4} - 2090128 T^{6} + 107021414 T^{8} - 2090128 p^{2} T^{10} + 28156 p^{4} T^{12} - 240 p^{6} T^{14} + p^{8} T^{16} \)
47 \( 1 - 312 T^{2} + 44912 T^{4} - 3894440 T^{6} + 222698718 T^{8} - 3894440 p^{2} T^{10} + 44912 p^{4} T^{12} - 312 p^{6} T^{14} + p^{8} T^{16} \)
53 \( 1 - 291 T^{2} + 41426 T^{4} - 70705 p T^{6} + 235496250 T^{8} - 70705 p^{3} T^{10} + 41426 p^{4} T^{12} - 291 p^{6} T^{14} + p^{8} T^{16} \)
59 \( ( 1 - 2 T + 150 T^{2} - 318 T^{3} + 11810 T^{4} - 318 p T^{5} + 150 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
61 \( ( 1 - 11 T + 222 T^{2} - 1885 T^{3} + 19674 T^{4} - 1885 p T^{5} + 222 p^{2} T^{6} - 11 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
67 \( 1 - 344 T^{2} + 53948 T^{4} - 5334440 T^{6} + 397158054 T^{8} - 5334440 p^{2} T^{10} + 53948 p^{4} T^{12} - 344 p^{6} T^{14} + p^{8} T^{16} \)
71 \( ( 1 + 15 T + 254 T^{2} + 2665 T^{3} + 27242 T^{4} + 2665 p T^{5} + 254 p^{2} T^{6} + 15 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
73 \( 1 - 460 T^{2} + 98468 T^{4} - 12885876 T^{6} + 1132966134 T^{8} - 12885876 p^{2} T^{10} + 98468 p^{4} T^{12} - 460 p^{6} T^{14} + p^{8} T^{16} \)
79 \( ( 1 + T + 140 T^{2} + 185 T^{3} + 16966 T^{4} + 185 p T^{5} + 140 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} )^{2} \)
83 \( 1 - 200 T^{2} + 27440 T^{4} - 3205848 T^{6} + 295212222 T^{8} - 3205848 p^{2} T^{10} + 27440 p^{4} T^{12} - 200 p^{6} T^{14} + p^{8} T^{16} \)
89 \( ( 1 + 9 T + 220 T^{2} + 2025 T^{3} + 23586 T^{4} + 2025 p T^{5} + 220 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
97 \( 1 - 231 T^{2} + 32558 T^{4} - 4062681 T^{6} + 461863330 T^{8} - 4062681 p^{2} T^{10} + 32558 p^{4} T^{12} - 231 p^{6} T^{14} + p^{8} T^{16} \)
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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

−3.69205776905851028041805369866, −3.58341198009767310795105480982, −3.32873240645577672483565968809, −3.18405829302524749912872217475, −3.09211367868720781543694887488, −3.08404297338018401387771812088, −2.74335704516143798182952387661, −2.72174839808971702558198421322, −2.71039585267873763470712031107, −2.64791759845230896278822645769, −2.36248764995748573307530239664, −2.24152533918423283725350812202, −2.16449439003086242632038018217, −2.10417263479008669087377474690, −2.10065829592837118655949822045, −1.70991079803197642200967126100, −1.39390351584221320972669031704, −1.34662723421781216754737869886, −1.33932172236459579893969008226, −1.03921646776115361471717209972, −0.950971254676633296051944658486, −0.930672807459434067782690129448, −0.24940741011716047740008863866, −0.22447652391590633429317048679, −0.21652667433679680995345736458, 0.21652667433679680995345736458, 0.22447652391590633429317048679, 0.24940741011716047740008863866, 0.930672807459434067782690129448, 0.950971254676633296051944658486, 1.03921646776115361471717209972, 1.33932172236459579893969008226, 1.34662723421781216754737869886, 1.39390351584221320972669031704, 1.70991079803197642200967126100, 2.10065829592837118655949822045, 2.10417263479008669087377474690, 2.16449439003086242632038018217, 2.24152533918423283725350812202, 2.36248764995748573307530239664, 2.64791759845230896278822645769, 2.71039585267873763470712031107, 2.72174839808971702558198421322, 2.74335704516143798182952387661, 3.08404297338018401387771812088, 3.09211367868720781543694887488, 3.18405829302524749912872217475, 3.32873240645577672483565968809, 3.58341198009767310795105480982, 3.69205776905851028041805369866

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

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